TITLE: Field property proof
QUESTION [0 upvotes]: I have tried quite a lot of "turn and twists" but I just can`t get around it. Here it is :
if x > y then x > z > y

for all $x,y$ in the partially ordered field $K$ exists such $z \in K$. For the real numbers f. e. that is perfectly clear but here I seem to believe that I don`t have enough info over $K$. Any thoughts, hints or suggestions on how to proceed?

REPLY [1 votes]: Put $z = \frac{x+y}{2}$ (the field can't have characteristic $2$ except in the trivial case when $x > y$ is always false). Then $x > z > y$.


TITLE: Integral behavior with small o notation
QUESTION [1 upvotes]: I would like to prove this simple result that seems true to me for an eigenvalue problem I am solving. This result would help me analyze the solutions to the resolvent equation.
Let $f$ be a continuous function such that $f(0)=0$ and let $k>1$ be real. Consider the integral
$$
I\equiv \int_x^1 \frac{f(x')}{x'^k}dx'\text{ for }x>0.
$$
I would like to prove that
$$
f\sim \mathcal{o}\!\left(x^{1-k}\right)\text{ as }x\rightarrow 0^+,\text{ i.e. }\lim_{x\to 0^+}x^{k-1}I(x)=0.
$$
I think it is true. I just don't know what to use exactly to prove it. Of course, I can be wrong, in which case a counter example is most welcome.

REPLY [1 votes]: Your result is true, at least so long as $\lim_{x\to0^+}I(x)$ exists (I believe this is always the case since $f$ is continuous at $0$, but I don't have a proof off the top of my head).
To see this, it is easiest to split into two cases:
If $\lim_{x\to0^+}I(x)=a$ for some $a\in\mathbb{R}$, we find clearly that $\lim_{x\to0^+}x^{k-1}I(x)=0\cdot a=0$.
If on the other hand $\lim_{x\to0^+}I(x)=\pm\infty$ we may use L'Hôpital's rule.
Indeed, suppose $\lim_{x\to0^+}I(x)=\infty$ (the other case is achieved by multiplying by $-1$) and note that $I'(x)=-\frac{f(x)}{x^k}$.
Thus we get that
$$
 \lim_{x\to0^+}x^{k-1}I(x)
 =\lim_{x\to0^+}\frac{I(x)}{x^{1-k}}
 =\lim_{x\to0^+}\frac{I'(x)}{\frac{\mathrm{d}}{\mathrm{d}x}[x^{1-k}]}
 =\lim_{x\to0^+}\frac{-f(x)}{x^k(1-k)x^{-k}}
 =0.
$$
EDIT: It is, in fact not always the case that $\lim_{x\to0^+}I(x)$ exists just because $f$ is continuous,
take e.g. the function $f(x)=x\sin(1/x)$ for $x\neq0$ and $f(0)=0$.
However the result above is true even if $I(x)$ is divergent at $0$ - simply let $f^+$ and $f^-$ denote the positive and negative parts of $f$, respectively and write $I(x)=I^+(x)-I^-(x)$, where $I^+(x)=\int_x^1\frac{f^+(x')}{x'^k}\mathrm{d}x'$ and similarly for $I^-$.
These will have a limit as $x\to0^+$ as they are monotone, and hence the above argument works.


text:
\begin{document}

\title{Performance Analysis of Backscatter Communication Systems with Non-orthogonal Multiple Access in Nakagami Fading Channels}

 \author{Ahsan~Waleed~Nazar,
         Syed~Ali~Hassan,~\IEEEmembership{Senior~Member,~IEEE,}
         Haejoon~Jung,~\IEEEmembership{Member,~IEEE,}
         Aamir~Mahmood,~\IEEEmembership{Senior~Member,~IEEE,}
         and~Mikael~Gidlund,~\IEEEmembership{Senior~Member,~IEEE}

\thanks{A. W. Nazar and  S. A. Hassan are with the  School of Electrical Engineering and Computer  Science~(SEECS), National University of Sciences and Technology~(NUST), Islamabad, Pakistan. (e-mail: anazar.msee23mcs@students.mcs.edu.pk, ali.hassan@seecs.edu.pk).}

\thanks{H. Jung is with the Department of Information and Telecommunication Engineering, Incheon National University, Incheon  22012, Korea (e-mail: haejoonjung@inu.ac.kr).}

\thanks{A. Mahmood and M. Gidlund are with the Department of Information Systems and Technology, Mid Sweden University, Sweden (emails: aamir.mahmood@miun.se, mikael.gidlund@miun.se).}}

\maketitle
\begin{abstract}
Backscatter communication (BackCom) has been emerging as a prospective candidate in tackling lifetime management problems for massively deployed Internet-of-Things devices, which suffer from battery-related issues, i.e., replacements, charging, and recycling. This passive sensing approach allows a backscatter sensor node (BSN) to transmit information by reflecting the incident signal from a carrier emitter without initiating its transmission. To multiplex multiple BSNs, power-domain non-orthogonal multiple access (NOMA), which is a prime candidate for multiple access in beyond 5G systems, is fully exploited in this work. Recently, considerable attention has been devoted to the NOMA-aided BackCom networks in the context of outage probabilities and system throughput. However, the closed-form expressions of bit error rate (BER) for such a system have not been studied. In this paper, we present the design and analysis of a NOMA enhanced bistatic BackCom system for a battery-less smart communication paradigm. Specifically, we derive the closed-form BER expressions for a cluster of two devices in a bistatic BackCom system employing NOMA with imperfect successive interference cancellation under Nakagami-$m$ fading channel. The obtained expressions are utilized to evaluate the reflection coefficients of devices needed for the most favorable system performance. Our results also show that NOMA-BackCom achieves better data throughput compared to the orthogonal multiple access-time domain multiple access schemes (OMA-TDMA).

\end{abstract}

\begin{IEEEkeywords}
 IoT, NOMA, Nakagami-$m$ fading, backscatter communication, bit error rate.
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\section{Introduction}

\textcolor{black}{Power-efficient wireless connectivity is at the heart of the massive proliferation of Internet-of-thing (IoT) networks for wide-scale and fine-grained sensing and control in various domains.  For instance, IoT networks of constrained battery-operated devices are attractive for applications in both the consumer and industrial domains, e.g., smart home/city/agriculture, e-health, building/industrial- automation, and vehicular/aerial networks \cite{dawy_toward_2017}. This pervasiveness and usability of IoT connectivity are equally reflected in currently connected IoT devices, which will further be reaching 24.5 billion devices by 2025 \cite{jejdling_ericsson_2020}. In this expected scenario, managing the network lifetime becomes critical, especially when the conventional battery-based solutions are not viable due to the high cost of battery replacements and recycling concerns. In particular, battery recharging/replacement is challenging in cases where most of the sensors are hidden (e.g., inside concrete walls) or deployed in an inaccessible environment (e.g., under cultivated land). As a result, various energy harvesting techniques are under investigation to overcome this challenge \cite{xu_internet_2014}.}

\textcolor{black}{Recently, for massive IoT networks, backscatter communication (BackCom) has emerged as one of the promising radio-frequency (RF) energy harvesting techniques, which enables communication without battery backup \cite{niyato_wireless_2017} \cite{Bletsas2018}.}
The low-complexity and low-power BackCom technique allows backscatter sensor nodes (BSNs) to communicate with a backscatter sensor reader (BSR) by passive reflection from a carrier emitter (CE) and modulation without requiring any active RF transmission component.

This backscatter is made possible because of the intentional impedance mismatch at the antenna input which results in different reflection coefficients \cite{lu_ambient_2018}. Data encoding for backscatter over an incident wave is carried out by varying the reflection coefficients at the antenna input side. This backscatter approach is vastly different from the generally applied wireless harvesting approach where nodes first harvest energy to perform active RF transmission.
Thus, transmissions in a backscatter communication consume orders of magnitude less energy than a typical radio and the absence of an active RF component in BSNs enables simpler and low complexity circuits. Therefore, the BackCom approach is most suitable for IoT networks \cite{liu_next_2019}.

The backscatter technique implementation for passive IoTs has been limited due to its inherent short-range communication. Recently, the BackCom system has been suggested for overcoming this limitation using bistatic architectures \cite{Kimionis2014},
\cite{Alevizos2018}. In bisatic architecture, the CE and BSR are not co-located, therefore, it allows setting up a more flexible network topology which can also mitigate the near-far effect.
Non-orthogonal multiple access (NOMA), because of its low latency and high spectral efficiency, is the ideal candidate to support a large number of IoT nodes in a BackCom uplink communication system \cite{ding_survey_2017}. Recently, NOMA-aided BackCom has proven to be a key technology for collating data from multiple BSNs \cite{Basnayake2020},
\cite{psomas_backscatter_2017}. In \cite{guo_design_2018}, the authors investigated the performance of a NOMA-enhanced BackCom system, and the significance of adopting NOMA with the BackCom system was demonstrated by analyzing the average number of successfully decoded bits. Similarly, the authors in \cite{zeb_noma_2019} and \cite{Le2019} evaluated the performance of NOMA-aided BackCom network in terms of outage probability and system throughput. The authors in \cite{Yang2019} and \cite{Liao2020} studied the problem of resource allocation in NOMA-enhanced BackCom networks. In \cite{Zhang2019}, the authors analyzed the outage probabilities and the ergodic rate for a symbiotic system that integrates cellular NOMA and ambient BackCom in an IoT network. The authors in \cite{Lyu2017} studied the optimal time allocation policies of a power station-powered BackCom system for a hybrid NOMA-TDMA scheme. Relay assisted BackCom system has been studied in \cite{Lyu2019}. In \cite{Farajzadeh2019}, the authors proposed a NOMA-BackCom application in unmanned aerial vehicle (UAV) based data collection with optimized UAV altitude and trajectory.

Based on the aforementioned literature survey, to the best of authors' knowledge, there is no reported work that carries out the BER performance analysis of uplink NOMA in a BackCom system, which is an essential component in system design and optimization. In this work, we consider a NOMA-enhanced bistatic backscatter uplink communication under the Nakagami-$m$ fading channel where one reader serves a cluster of randomly deployed BSNs. To maintain low NOMA decoding complexity, BSNs are usually divided into a cluster of two users. The reflection coefficients of the BSNs are set different from each other to make the wireless channel distinct to better exploit power-domain NOMA.
The proposed scheme results in an increase in effective non-erroneous transmitted bits as compared to an OMA scheme. The main contributions of this paper are summarized as follows:

\begin{itemize}
  \item We propose a NOMA enhanced bistatic BackCom system for battery-less smart communication among BSNs, where the reflection coefficients of BSNs can be manipulated to achieve better system performance in terms of effective data bits.
  \item The BER performance of a NOMA enhanced bistatic BackCom system, impaired by a Nakagami-$m$ fading channel, with imperfect SIC is considered, where approximate closed-form analytical BER expressions of binary phase-shift keying (BPSK) are derived for two BSNs. The derived BER expressions are verified by Monte Carlo simulations for various reflection coefficients.
  \item The probability density function (PDF) of the difference of two independent and non-identically distributed (i.n.i.d) Nakagami-$m$ distributions has not been derived in literature before. It is shown with the help of some statistical approaches such as the moment matching method that it follows a normal distribution and we derive the average BER expressions under such a scenario.
  \item The increase in effective non-erroneous transmitted bits over a large period in a NOMA-BackCom scheme is compared with an OMA scheme for a variety of  reflection coefficient conditions. It is shown that NOMA-BackCom outperforms the OMA-TDMA scheme by multiplexing the BSNs.
  \item Our results suggest that at relatively higher transmit SNRs, system performance can be greatly improved by setting suitable reflection coefficients rather than an increment in transmit power of the CE.

\end{itemize}

The rest of the paper is organized as follows. In Sec. II, the system model of the NOMA-aided BackCom system is briefly described. It is followed by the derivation of closed-form BER expressions of BPSK constellation for two BSNs in Sec. III. Numerical and simulation results are shown in Sec. IV. Finally, conclusions are presented in Sec. V.

\section{System Model}
\subsection{BackCom Model}
We consider a NOMA-aided bistatic BackCom system (BBCS) consisting of a CE, multiple BSNs, and a BSR as shown in Fig. \ref{fig:System_model}. In practice, BSNs are usually multiplexed into different clusters of two or three users to maintain low decoding complexity and meet timing constraints. Moreover, CEs are placed closer to the BSNs in the field to mitigate the doubly near-far effect. The BSNs do not possess any active RF transmission source. The CE transmits a sinusoidal continuous wave (CW) carrier signal which is intelligently reflected by BSNs to aid in communication with BSR\@. The CW signal is transmitted by CEs all the time, whereas BSNs operate in two states, namely the active state and energy harvesting state.

\begin{figure}[t]
  \centering
  \includegraphics[width= 0.9\columnwidth]{System_model.pdf}
  \caption{Illustration of uplink NOMA-aided BBCS.}
  \label{fig:System_model}

\end{figure}

In the \emph{active state}, each cluster of BSNs backscatters the CW signal to transmit its sensor data to the BSR\@. The BSR receives the signal and recovers the information from BSNs. BSNs are capable of reflecting the incident CW signal with altered phase and/or magnitude by terminating its antenna between two load impedances. This is typically carried out with the help of an RF transistor. Ideally, by switching the transistor on, the antenna is short-circuited and the incident wave is reflected with a phase change of 180\degree. Alternatively, by switching the transistor off, the antenna is open-circuited and the incident wave is reflected with no phase change (i.e., zero degree). By changing the value of load impedance from short and open-circuit conditions, the magnitude of the reflected wave can also be changed alongside phase.

In the \emph{energy harvesting state}, the BSNs do not reflect the incident CW signal, but only harvest the energy from it. The harvested energy is stored in a battery and is used to power the circuitry (including micro-controller and RF transistor) and support sensing operations. We assume that the cluster which is not in an active state is in an energy harvesting state or vice versa. Furthermore, the stored energy can be used to sustain long term operations of BSNs\@.

As the modulation of BSNs is achieved by switching the antenna loads between \(M\) load impedances, it corresponds to the power reflection coefficients, \(\Gamma_n\), \(n =\) \{0\,,\,.\,.\,.\,, \( M-1\)\}. Keeping in view the complexity and energy constraints of the low-power BSNs, we consider BPSK modulation in this work. Therefore, we only consider two values of power reflection coefficient \{\(\Gamma_0\), \(\Gamma_1\)\} corresponding to two load impedances. The switching of RF transistor between the two states for the BPSK modulation is performed with the help of a micro-controller. A typical BSN (see Fig. \ref{fig:tag}) circuit consists of an antenna, transmitter, receiver, energy harvester, variable impedances, micro-controller, sensor, RF transistor switch, and a battery.

\begin{figure}[ht]
  \centering
  \includegraphics[width=\columnwidth]{tag.pdf}
  \caption{Illustration of the internal structure of a BSN.}
  \label{fig:tag}

\end{figure}

\subsection{Channel~Model}
We consider an uplink transmission scenario in which each cluster of two BSNs is served by a single CE and BSR\@. Both BSNs and BSRs are equipped with single antennas. As shown in Fig. \ref{fig:System_model}, \(h_{f,i}\) is used to denote the channel coefficient of the forward link between the CE and \(i\)th BSN, for \(\,i  \in\{1,2\}\), whereas \(h_{b,i}\) is used to denote the  channel coefficient of the backscatter link between \(i\)th BSN and BSR.

For the \emph{forward channel}, path loss only propagation effects are considered in \(h_{f,i}\). This fading-free channel model is a reasonable channel assumption because of the proximity and strong line-of-sight (LoS) link between CE and BSNs\footnote{A fading impaired channel model for forward link is left as a future study at the moment.}. Combined with our BackCom model and given that the transmit power of the CE is \(P_{T}\), the power required at a BSN for forward channel is given by \(P_{T}\,\varGamma_{i}\,|h_{f,i}|^2\).
In the \emph{backscatter channel}, we assume that the channel coefficient \(h_{b,i}\) follow independent and non-identically distributed (i.n.i.d) Nakagami-\(m\) fading which can model both LoS and non-LoS conditions. The zero mean additive white Gaussian noise (AWGN), \(w\), with variance \(N_0\), i.e., \(w\sim \mathcal{C}\mathcal{N}\)(0, \(N_0\)) is also considered in the system.

\subsection{NOMA Scheme}
We employ a power-domain NOMA (PD-NOMA) uplink scheme in the BackCom system. PD-NOMA operates by multiplexing users with relatively large channel gain differences over same time/frequency slot, thereby improving spectral efficiency. During IoT deployment, sometimes grouping IoTs with nearly balanced power differences is inevitable. However, in IoT scenarios where conventional PD-NOMA is unusable, its BackCom variant can still function by manipulating the power reflection coefficients of the BSNs to generate acceptable channel differences. In a cluster of two BSNs, a training phase is required by the BackCom to differentiate the BSNs into a weak or strong user. It works as follows: each BSN is distinguishable by its unique ID which is known to the BSR. The BSR broadcasts a pilot signal with the unique ID for each BSN in two training time slots. The BSNs backscatter the signal with the same power reflection coefficient after receiving its unique ID. Now, the BSR can obtain the instantaneous channel state information (CSI) and can classify the BSNs into weak/strong pair. The BSNs can then set its power reflection coefficients corresponding to the received instantaneous CSI by the BSR.

For a single cluster, the received signal at the BSR can be written as

\begin{equation}\label{eq:composite}
  y = \sqrt{P_T \varGamma_1} h_{f,1} h_{b,1} x_1 + \sqrt{P_T \varGamma_2} h_{f,2} h_{b,2}  x_2 + w,
\end{equation}
where \(x_i\) is the BPSK modulated information signal of $i$th nodes. As mentioned earlier, the forward channels \(h_{f,i}\)s are assumed to be dominated by a deterministic path loss model only, whereby their effects are compensated in the transmit SNR of both BSNs and are thus omitted in further analysis. In order to decode the signals transmitted by the BSNs, successive interference cancellation (SIC) process is implemented at the BSR, where error propagation may happen. Without the loss of generality, it assumed that the first BSN has higher channel gain than the second BSN, i.e., \(|h_{b,1}|^2 > |h_{b,2}|^2 \). To detect the weaker signal in a SIC scheme, the stronger  signal should be detected first and scaled, then subtracted from the aggregate received signal. Therefore, the optimal decoding order is in the order of decreasing channel gains. Considering this, the signal from BSN-1, $u_1$, is decoded firstly by treating BSN-2, $u_2$, as inter-user interference (IUI) at the BSR. Thus the maximum likelihood detector (MLD) of $u_1$, given that the channel gains are estimated perfectly at BSR, can be described by \cite{proakis2007digital}

\begin{equation}\label{eq:MLD}
  \widehat{x}_1  =
  \underset{\widetilde{x}_1 \in \mathcal{S}}{\operatorname{arg\,min}} \bigg| y- \sqrt{P_T \varGamma_1} h_{b,1} \widetilde{x}_1\bigg|^2,
\end{equation}
where $\widehat{x}_1 $ is the estimated data symbol, $\mathcal{S}$ is the set of all possible constellation points for \(u_1\), and $\widetilde{x}_1$ is the set of all possible trial values for \(x_1\).

Next, $u_2$ signal is decoded after subtracting the detected $u_1$ symbol from received composite signal \(y\). If $u_1$'s signal is decoded correctly then no IUI is faced in decoding of $u_2$, otherwise, there will be error propagation from $u_1$ while decoding $u_2$. The detector for $u_2$ can be expressed as

\begin{equation}\label{eq:MLD1}
  \widehat{x}_2  =
  \underset{\widetilde{x}_2 \in \mathcal{S}}{\operatorname{arg\,min}} \bigg| \bigg(y- \sqrt{P_T \varGamma_1} h_{b,1} \widehat{x}_1 \bigg) - \sqrt{P_T \varGamma_2} h_{b,2} \widetilde{x}_2 \bigg|^2.
\end{equation}

In the next section, BER expressions are derived for PD-NOMA-BackCom system with a cluster of two BSNs, assuming equiprobable symbols. The same approach may be applied for a higher order phase shift keying (PSK) modulation, however, the derivation becomes impractical when a large number of BSNs are multiplexed together.

\section{NOMA-BackCom BER Performance Analysis}

As noted from (\ref{eq:composite}), the received symbol at BSR is a superposition of two BPSK symbols, therefore, it corresponds to a total of four constellation points as shown in Fig. \ref{fig:constellation}. Because \(|h_{b,1}|^2 > |h_{b,2}|^2 \), the BSR detects \(u_1\) symbol first. Thereafter, it subtracts the decoded \(u_1\) symbol from the received signal and detects \(u_2\) symbol. Each constellation point in Fig. \ref{fig:constellation} is represented by two bits given by \(\{x_1, x_2\}\). Here, \(x_1\) is the BPSK bit of $u_1$ and \(x_2\) is the BPSK bit of $u_2$. In Fig. \ref{fig:constellation}, red diamond shows the original location of \(x_1\) without IUI from $u_2$. However, because of the presence of IUI from $u_2$, \(x_1\) is translated to two possible constellation points. The shaded block shows the two possible values that a particular $u_1$ bit \(x_1\) may take due to IUI. By modifying the values of reflection coefficients, the position of these constellation points can be changed, which will in turn affect the BER performance of the system.

\begin{figure}[ht]
  \centering
  \includegraphics[width=\columnwidth]{constellation.pdf}
  \caption{The received signal space diagram  of super-imposed BPSK symbols from two BSNs at the BSR, where \(|h_{b,1}|^2 > |h_{b,2}|^2 \).}
  \label{fig:constellation}

\end{figure}

\subsection{BER of the First User}
The detection of the first user is performed according to (\ref{eq:MLD}), therefore, no SIC is required in the process. An error will be made in the detection process if $\widehat{x}_1\neq  x_1 $ and its probability is given by $P_{e}(u_1)$. The probability of error for $u_1$ depends on the decision boundary distance of $x_1$ and it is the sum of error probabilities of each possible symbol multiplied with the prior probability. Because of IUI, there are four possible cases in which a bit $x_1$ can be decoded incorrectly. When \(u_1^{(0)}\) and \(u_2^{(0)}\) are sent, where \(u_i^{(y)}\) represents that user $i$ transmits a bit $y$, the decision boundary of $x_1$, represented by the red line, is at a distance of \( \sqrt{\varepsilon_1 \Gamma_1}|h_{b,1}| + \sqrt{\varepsilon_2 \Gamma_2 }|h_{b,2}| \) from the constellation point \{0,\! 0\}, where $\varepsilon_i$ denote the symbol energy of user $i$. Therefore, bit $x_1$ will be decoded incorrectly if in-phase component of $w$ exceeds the sum of $u_1^{(0)}$ and $u_2^{(0)}$ i.e., $w \geq  \sqrt{\varepsilon_1 \Gamma_1}|h_{b,1}| + \sqrt{\varepsilon_2 \Gamma_2 }|h_{b,2}| $. Similarly, following the same procedure for each symbol, the error is calculated and multiplied by prior probability. Consequently, the error probability for $u_1$, considering even symmetry in constellation diagram, is given by

\begin{equation}\label{eq:P1e}
  \begin{split}
    P_{e}(u_1)  =  \frac{1}{4} \left[  \mathbb{P}(|w|\geq \Upsilon_1  + \Upsilon_2  )  \right.
    + \left. \mathbb{P}(|w|\geq \Upsilon_1 - \Upsilon_2  )  \right],
  \end{split}
\end{equation}
where $\mathbb{P}(x)$ denotes the probability of event $x$, and $\Upsilon_i=\sqrt{\varepsilon_i \Gamma_i}|h_{b,i}|$.
The expression (\ref{eq:P1e}) can be represented using Gaussian $\mathcal{Q} $ function as

\begin{equation}\label{eq:P1e_Q}
  \begin{split}
    P_{e}(u_1)  =  \frac{1}{2} \left[ \mathcal{Q}\left(Y \right) + \mathcal{Q}\left(Z \right)  \right],
  \end{split}
\end{equation}

where $Y$ and $Z$ are random variables (RVs), defined as

\begin{equation}\label{eq:Y_Z}
  \begin{split}
    Y = \frac{\Upsilon_1}{\sqrt{N_0/2}} + \frac{\Upsilon_2}{\sqrt{N_0/2}}, \\
    Z = \frac{\Upsilon_1}{\sqrt{N_0/2}} - \frac{\Upsilon_2}{\sqrt{N_0/2}},
  \end{split}
\end{equation}
and the Gaussian $\mathcal{Q} $ function is defined as

\begin{equation}\label{eq:Qfunc}
  \mathcal{Q}(x) = \frac{1}{2\pi} \int_{x}^{\infty}\exp\bigg(\!-\frac{u^2}{2}\bigg)  \,\mathrm{d}u.

\end{equation}

The average BER for BSN-1, denoted as $\overline{P_{e}(u_1)}$, is evaluated by averaging over the PDF of RV $Y$, $f_{Y}(y)$, and RV $Z$, $f_{Z}(z)$, where, $f_{Y}(y)$ and $f_{Z}(z)$ are the PDFs for the sum and difference of two i.n.i.d Nakagami-$m$ distributions, respectively. The average error probability of $u_1$ is

\begin{equation}\label{eq:P1e_Q1}
  \begin{split}
    \overline{P_{e}(u_1) }  =  \frac{1}{2} \left[ \int_{-\infty}^{\infty }   \mathcal{Q}\left(Y\right)\! f_{Y}(y) \,\mathrm{d}y + \right.
    \left. \int_{-\infty}^{\infty }  \mathcal{Q}\left(Z \right)\! f_{Z}(z) \,\mathrm{d}z \right].
  \end{split}
\end{equation}

The PDF $f_{Y}(y)$ has been derived in \cite{Annamalai2000} and \cite{Rahman2010} in the form of Appell hypergeometric function of the second kind and Lauricella multivariable hypergeometric function, respectively. However, a closed-form solution of (\ref{eq:P1e_Q1}) is prohibited using expressions in \cite{Annamalai2000} and \cite{Rahman2010}, rendering the use of numerical evaluation. Therefore, the PDF of $Y$ in this work is approximated with another distribution, complying with the features of $Y$ as shown in the following lemma.

\emph{Lemma 1:} The distribution of the sum of two i.n.i.d Nakagami-$m_i$ RVs, $Y_1$ and $Y_2$, with parameters $m_i$ and $\Omega_i$, where \(\,i  \in\{1,2\}\),  can be approximated by another Nakagami-$m_{R_1}$ RV $R_1$, i.e., $R_1=Y_1+Y_2$, with fading parameter $m_{R_1}$ and average power $\Omega_{R_1}$, with PDF as

\begin{equation}\label{eq:PDF_nak}
  f_{R_1}(r) =  \frac{2m_{R_1}^{m_{R_1}}r^{2m_{R_1}-1}}{\Omega_{R_1}^{m_{R_1}}\Gamma (m_{R_1})}\text{exp}(-\frac{m_{R_1}r^2}{\Omega_{R_1}} ) ,\quad r_1> 0,
\end{equation}
where $\Omega_{R_1}$ and $m_{R_1}$ are defined as \cite{NAKAGAMI19603}

\begin{equation}\label{eq:m_O}
  \begin{split}
    \Omega_{R_1} &= \mathbb{E}[{R_1}^2],\\
    m_{R_1} = \frac{(\mathbb{E}[{R_1}^2])^2}{Var[{R_1}^2]} &= \frac{\Omega_{R_1}^2 }{\mathbb{E}[{R_1}^4]-\Omega_{R_1}^2},
  \end{split}
\end{equation}
where $\Gamma(.)$ is the gamma function, and $\mathbb{E}[.]$ and $Var[.]$ denote the expectation and variance operators, respectively.

\begin{comment}
  \begin{figure*}[h]
  \centering
  \subfloat[$m_1=1, \Omega_1 =1,m_2=1, \Omega_2 =1$]{\includegraphics[width= 2.5in]{ecdf1111}\label{fig:ecdf1}}
  \subfloat[$m_1=2, \Omega_1 =1,m_2=1, \Omega_2 =1$]{\includegraphics[width= 2.5in]{ecdf2111}\label{fig:ecdf2}}
  \subfloat[$m_1=3, \Omega_1 =1,m_2=3, \Omega_2 =0.5$]{\includegraphics[width= 2.5in]{ecdf313.5}\label{fig:ecdf3}}
  \caption{Plot of empirical CDF vs theoretical CDF obtained using MOM for the sum of two i.n.i.d Nakagami-$m$ distributions}
  \label{fig:ecdf}
\end{figure*}
\end{comment}

\emph{Proof:} A two-step process is followed to approximate the distribution of the sum of two RVs $Y_1$ and $Y_2$ to another distribution. In the first step, the method of moments (MoM) approach is utilized to match the moments of RV $Y=Y_1+Y_2$ to that of Nakagami-$m_{R_1}$ RV, ${R_1}$. Nakagami-$m$ distribution is based on two parameters, i.e., fading parameter $m$ and shape parameter $\Omega$. Therefore, we need to match the moments of the two parameters. From \cite{NAKAGAMI19603}, the $n$th moment of the Nakagami-$m_i$ distribution is given by

\begin{equation}\label{eq:nmoments}
  \mathbb{E}[Y_i^n] = \frac{\Gamma(m_i+\frac{n}{2})}{\Gamma(m_i)} \bigg(\frac{\Omega_i}{m_i}\bigg)^{\frac{n}{2}},
\end{equation}
where $m_i$ is the fading parameter of each RV $Y_i$, $\Omega_{1}=\Omega_{Y_1}=2 \Upsilon_1^2/N_0$ and $\Omega_{2}=\Omega_{Y_2}=2 \Upsilon_2^2/N_0$. Using the multinomial theorem assuming independence among RVs, $Y_1$, $Y_2$, and (\ref{eq:nmoments}), the second moment of the sum RV $Y=Y_1+Y_2$ is matched with the second moment of RV $R_1$, i.e.,

\begin{align}\label{eq:2moment}

    &\Omega_{R_1}={\mathbb{E}[Y^2]} = \mathbb{E}[Y_1^2] +\mathbb{E}[Y_2^2]+2\mathbb{E}[Y_1]\mathbb{E}[Y_2]\nonumber\\
    &= \Omega_1 + \Omega_2+ 2 \frac{\Gamma(m_1+\frac{1}{2})}{\Gamma(m_1)} \bigg(\frac{\Omega_1}{m_1}\bigg)^{\frac{1}{2}} \frac{\Gamma(m_2+\frac{1}{2})}{\Gamma(m_2)} \bigg(\frac{\Omega_2}{m_2}\bigg)^{\frac{1}{2}} .

\end{align}

Now we need to find the value of $m_{R_1}$. To this objective, we note from (\ref{eq:m_O}) that $\mathbb{E}[{R_1}^4]$ is required. Again using the multinomial theorem for ${R_1}=Y_1+Y_2$, $\mathbb{E}[{R_1}^4]$ can be evaluated as

\begin{equation}\label{eq:4moment}
    \mathbb{E}[{R_1}^4] = \mathbb{E}[Y_1^4] + 4\mathbb{E}[Y_1^3]\mathbb{E}[Y_2] +6\mathbb{E}[Y_1^2]\mathbb{E}[Y_2^2]
     +4\mathbb{E}[Y_1]\mathbb{E}[Y_2^3] +\mathbb{E}[Y_2^4].
\end{equation}

Then, from (\ref{eq:m_O}) and (\ref{eq:2moment}), value of $m_{R_1}$ can easily be found.

In the second step, the Kolmogorov-Smirnov (K-S) test is used for goodness-of-fit (see Appendix A).
Fig. \ref{fig:ECDF}\ref{sub@fig:ECDF_Nakagami} shows the plot of empirical CDF and approximated theoretical CDF.

\begin{figure}[!t]
  \centering
  \vspace{-10pt}
  \subfloat[]{\includegraphics[width= 3.35in]{ECDF_Nakagami_1.pdf}\label{fig:ECDF_Nakagami}}
  \subfloat[]{\includegraphics[width= 3.35in]{ECDF_Normal_1.pdf}\label{fig:ECDF_Normal}}
  \caption{Empirical CDF vs theoretical CDF obtained using MOM with two i.n.i.d Nakagami-$m$ distributions; (a) the sum of the two, (b) the difference of two.}

  \label{fig:ECDF}
\end{figure}

The PDF $f_{Z}(z)$ for the difference of two i.n.i.d Nakagami-$m$ distributions has not been derived previously. Therefore, we approximate it with another closely matching distribution using the following lemma.

\emph{Lemma 2:} The distribution of the difference of two i.n.i.d Nakagami-$m_i$ RVs, $Z_1$ and $Z_2$, with parameters $m_i$ and $\Omega_i$, where \(\,i  \in\{1,2\}\), can be approximated by a Gaussian (normal) RV, ${W_1}=Z_1-Z_2$, with mean $\mu_{W_1}$ and variance $\sigma^2_{W_1}$, with PDF

\begin{equation}\label{eq:normal_PDF}
  f_{W_1}(w) = \frac{1}{\sqrt{2\pi\sigma_{W_1}^2}}\exp\bigg(-\frac{(w-\mu_{W_1})^2}{2\sigma^2_{W_1}}\bigg),
\end{equation}
where the values of $\mu_{W_1}$ and $\sigma^2_{W_1}$ are given in (\ref{normal_1_mom}) and (\ref{normal_2_mom}).

\emph{Proof:}  A similar two-step process as used in Lemma 1, is used to approximate the distribution of the difference of two RVs $Z_1$ and $Z_2$ to another distribution. Firstly, MoM is applied to match the moments of RV $Z=Z_1-Z_2$ to moments of the normal RV, ${W_1}$. The first two moments of ${W_1}$ are

 \begin{equation}\label{normal_moments}
   \begin{split}
     \mathbb{E}[{W_1}] &= \mu_{W_1},\\
     \mathbb{E}[{W_1}^2] &= \mu_{W_1}^2+\sigma_{W_1}^2.
   \end{split}
 \end{equation}

Following the algorithm of moment matching, using multinomial theorem, noting independence among RVs, $Z_1$ and $Z_2$, and using (\ref{eq:nmoments}), the mean $\mu_{W_1}$ of the normal RV, ${W_1}$, can be evaluated by matching the first moment of the difference RV $Z=Z_1-Z_2$  with RV ${W_1}$ as
\begin{align}\label{normal_1_mom}

    \mu_{W_1} &= \mathbb{E}[Z] = \mathbb{E}[Z_1] -\mathbb{E}[Z_2]\nonumber\\
    &= \frac{\Gamma(m_1+\frac{1}{2})}{\Gamma(m_1)} \bigg(\frac{\Omega_1}{m_1}\bigg)^{\frac{1}{2}} - \frac{\Gamma(m_2+\frac{1}{2})}{\Gamma(m_2)} \bigg(\frac{\Omega_2}{m_2}\bigg)^{\frac{1}{2}},

\end{align}
where $\Omega_{1}=\Omega_{Z_1}=2 \Upsilon_1^2/N_0$ and $\Omega_{2}=\Omega_{Z_2}=2 \Upsilon_2^2/N_0$. Now we match the second moments to obtain variance $\sigma_{W_1}^2$

\begin{align}\label{normal_2_mom}

    \mu_{W_1}^2+\sigma_{W_1}^2 &=\mathbb{E}[Z^2] = \mathbb{E}[Z_1^2] +\mathbb{E}[Z_2^2] -2\mathbb{E}[Z_1]\mathbb{E}[Z_2],\nonumber\\
    \sigma_{W_1}^2 &= \mathbb{E}[Z^2] - \mu_{W_1}^2,

\end{align}
where $\mathbb{E}[Z_i^n]$ is taken from (\ref{eq:nmoments}).

In the second step, the Kolmogorov-Smirnov (K-S) test is used to show that the normal distribution closely matches the distribution of the difference of two Nakagami-$m$ RVs. The K-S test is
has been performed in Appendix B.
 Fig. \ref{fig:ECDF}\ref{sub@fig:ECDF_Normal} shows the plot of empirical CDF and approximated theoretical CDF showing a close agreement.

Now from (\ref{eq:PDF_nak}) and (\ref{eq:normal_PDF}), a closed-form solution for the average BER can be evaluated as follows

\begin{equation}\label{eq:Pu1_avergae}
  \overline{P_{e}(u_1) } = \frac{1}{2} \Big[ \underbrace{\int_{-\infty}^{\infty }\!\!\!\!\! \mathcal{Q}(R) f_{R }(r ) \,\mathrm{d}r}_{\Phi}  + \underbrace{\int_{-\infty }^{\infty } \!\!\!\!\!  \mathcal{Q}\left(W \right)\! f_{W}(w) \,\mathrm{d}w}_{\Lambda }  \Big].
\end{equation}
The closed-form expression for $\Phi$ can be found by invoking the alternative form of the Q-function known as Craig's formula \cite{Simon2005} and applying moment generating function (MGF) for $f_{\gamma_R }(\gamma_r )$. It is given by

\begin{equation}\label{eq:Nak_MGF}
  \begin{split}
    \Phi(m,\overline{\gamma}) =
    \begin{cases}
      \frac{1}{2} \Bigg[ 1 - \Psi \big(\frac{\overline{\gamma}}{2m}\big) \sum\limits_{k = 0}^{m-1}\bigg(\frac{1-\Psi^2\big(\frac{\overline{\gamma}}{2m}\big)^k}{4}\bigg) \Bigg],
       \text{$m$ integer}\\
      \frac{1}{2\sqrt{\pi}}\frac{\sqrt{\frac{\overline{\gamma}}{2m}}}{(1+\frac{\overline{\gamma}}{2m})^{m+(1/2)}} \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m+1)} \times \\
      {}_{2}F_{1} \bigg( 1,m+\frac{1}{2};m+1;\frac{m}{m+\frac{\overline{\gamma}}{2m}}\bigg),
      \: \text{$m$ noninteger}
    \end{cases}
  \end{split}
\end{equation}
\begin{comment}
  \begin{equation}\label{eq:Nak_MGF}
  \begin{split}
    \Phi(m,\overline{\gamma}) =
    \begin{cases}
      \frac{1}{2} \Big[ 1 - \Psi \big(\frac{\overline{\gamma}}{2m}\big) \sum\limits_{k = 0}^{m-1}\bigg(\frac{1-\phi^2\big(\frac{\overline{\gamma}}{2m}\big)^k}{4}\bigg) \Big], \\
      \qquad \Psi\big(\frac{\overline{\gamma}}{2m}\big)\triangleq \sqrt{\frac{\overline{\gamma}/2}{m+\overline{\gamma}/2}},\\
       \text{$m$ integer}\\
      \frac{1}{2\sqrt{\pi}}\frac{\sqrt{\frac{\overline{\gamma}}{2m}}}{(1+\frac{\overline{\gamma}}{2m})^{m+(1/2)}} \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m+1)} \\
      \qquad \times {}_{2}F_{1} \big( 1,m+\frac{1}{2}:m+1;\frac{m}{m+\frac{\overline{\gamma}}{2m}}\big).\\
       \text{$m$ noninteger}
    \end{cases}
  \end{split}
\end{equation}
\end{comment}
where $\Psi\big(\frac{\overline{\gamma}}{2m}\big)\triangleq \sqrt{\frac{\overline{\gamma}/2}{m+\overline{\gamma}/2}}$ and ${}_{2}F_{1}$
is the Gauss hypergeometric function. Further, the closed-form expression for $\Lambda$ in (\ref{eq:Pu1_avergae}) is given by following Lemma.

\emph{Lemma 3:} The integral of the product of a Q function and a normal distribution with mean $\mu$ and variance $\sigma^2$ is

\begin{equation}\label{eq:normal_closed}
 \Lambda(\mu,\sigma^2) = \mathcal{Q}\bigg(\frac{\mu}{\sqrt{\sigma^2+1}}\bigg).
\end{equation}

\emph{Proof:} See Appendix C.

Finally by using (\ref{eq:Nak_MGF}) and (\ref{eq:normal_closed}) in (\ref{eq:Pu1_avergae}), the average probability of error for BSN-1 can be expressed as

\begin{equation}\label{eq:Pu1_average_closed}
  \overline{P_{(e)}(u_1) } = \frac{1}{2} \left[ \Phi(m_{R_1},\Omega_{R_1}) + \Lambda(\mu_{W_1},\sigma^2_{W_1})\right] .
 \end{equation}

\subsection{BER of the Second User}

To decode the second user, SIC process is implemented, whereby, BSR would initially detect the first user according to (\ref{eq:MLD}) and then from the reconstructed $\widehat{x}_1$ bit, compute,

\begin{equation}\label{MLD_2ndSIC}
  \begin{split}
    \widehat{x}_2= \underset{\widetilde{x}_2 \in \mathcal{S}}{\operatorname{arg\,min}} \bigg| y_{SIC} + w - \sqrt{P_T \varGamma_2} h_{b,2} \widetilde{x}_2\bigg|^2.
  \end{split}
\end{equation}

Therefore, if $\widehat{x}_1$ is detected correctly, i.e., $\widehat{x}_1= x_1$, then $y_{SIC} = \sqrt{P_T \varGamma_2} h_{b,2} x_2$ and it corresponds to an IUI-free decoding. On the other hand, if $\widehat{x}_1$ is detected incorrectly, i.e., $\widehat{x}_1\neq  x_1$, then $y_{SIC} = \sqrt{P_T \varGamma_1} h_{b,1}x_1 + \sqrt{P_T \varGamma_2}h_{b,2} x_2 - \sqrt{P_T \varGamma_1}h_{b,1} \widehat{x}_1 $ and the decoding decision for second user also depends on the value of reconstructed bit $ \widehat{x}_1$. Fig. \ref{fig:Constellation-I_II} shows the received signal diagram for the two aforementioned cases when the transmitted $u_1$ bit $x_1^{(1)}$ is decoded correctly or incorrectly. Therefore, these two cases should be handled differently.

\begin{figure}[ht]
  \centering
  \includegraphics[width=0.8\columnwidth]{Constellation-I_II}
  \caption{The received signal space diagram of BSN-2 for $x_1^{(1)}$ transmission when $y_{SIC}|_{\widehat{x}_1\neq  x_1}$ and $y_{SIC}|_{\widehat{x}_1=  x_1}$.}
  \label{fig:Constellation-I_II}

\end{figure}

\subsubsection{Case I}

We first consider the case that $u_1$ symbol has been detected correctly by BSR. The probability for correct decoding of $x_1$ bit is opposite to the one derived in (\ref{eq:P1e}), which gives the probability for incorrect decoding of $x_1$ bit. This will serve as the prior probability for case I. The error probability for $x_2$ bit is influenced by the decision boundary for detecting $u_2$ symbol as shown by the constellation points indicated by the black circles in Fig. \ref{fig:Constellation-I_II}. Hence, the probability of error for $u_2$ with correct $u_1$ decoding, considering both $x_1^{(0)}$ and $x_1^{(1)}$ bit transmission scenarios, is given by (\ref{eq:P_e_u2_I}) at the top of page.

The conditional expressions pertain to decoding the $x_1$ bit correctly and can be inferred from Fig. \ref{fig:constellation}. The first and second terms in (\ref{eq:P_e_u2_I}) correspond to $x_2^{(0)}$ and $x_2^{(1)}$ received constellation points, respectively.

\begin{figure*}[!t]
  \vspace{-10pt}
  \normalsize
  \setcounter{MYtempeqncnt}{\value{equation}}
  \setcounter{equation}{22}
  \begin{multline}
  \label{eq:P_e_u2_I}
  P_e^I(u_2) = \frac{1}{2} \left[ \mathbb{P}(w\leq \Upsilon_1 + \Upsilon_2 ) \times \mathbb{P}\big(w\geq   \Upsilon_2\bigm\vert w\leq \Upsilon_1 + \Upsilon_2 \big) +  \mathbb{P}(w\leq \Upsilon_1 - \Upsilon_2 ) \times \right. \\ \left.\mathbb{P}\big(w\leq -\Upsilon_2\bigm\vert w\leq \Upsilon_1 - \Upsilon_2 \big)\right].
  \end{multline}
  \vspace{-10pt}
  \begin{multline}\label{eq:P_e_u2_II}
  P_e^{II}(u_2) =  \frac{1}{2} \left[ \mathbb{P}(w\geq \Upsilon_1 + \Upsilon_2 ) \times  \mathbb{P}\big( w\geq 2\Upsilon_1 + \Upsilon_2 \bigm\vert w\geq \Upsilon_1 + \Upsilon_2 \big)+ \mathbb{P}(w\geq \Upsilon_1 - \Upsilon_2 )\times \right. \\ \left. \mathbb{P}\big( w\leq 2\Upsilon_1 - \Upsilon_2 \bigm\vert w\geq \Upsilon_1 - \Upsilon_2 \big) \right].
  \end{multline}
  \setcounter{equation}{\value{equation}}
  \hrulefill
  \vspace*{-4pt}
\end{figure*}

Using conditional probability law, (\ref{eq:P_e_u2_I}) can be rewritten as

\begin{equation}\label{eq:P_e_u2_I_simplified}
  P_e^I(u_2) = \frac{1}{2}\left[ \mathbb{P}(\Upsilon_2 \leq w \leq \Upsilon_1 + \Upsilon_2) + \mathbb{P}(\Upsilon_2 \leq w)\right].
\end{equation}

The  expression in (\ref{eq:P_e_u2_I_simplified}) can be  represented  using  Gaussian $\mathcal{Q}$ function as

\begin{equation}\label{eq:P_e_u2_I_Q}
  P_e^I(u_2) = \mathcal{Q}(\sqrt{\gamma_{\Upsilon_2}}) - \frac{1}{2}\mathcal{Q}(Y),
\end{equation}
where $Y$ is defined in (\ref{eq:Y_Z}) and $\gamma_{\Upsilon_2} = \frac{\Upsilon_2^2}{N_0/2}$.

\subsubsection{Case II}
This case considers the scenario when $u_1$ symbol has been detected incorrectly by BSR. The probability for incorrect decoding of $x_1$ bit is similar to the one derived in (\ref{eq:P1e}). This will serve as the prior probability for case II. Because of the IUI from $u_1$, there will be an error propagation of $2\Upsilon_1$ from the decision boundary to detect $x_2$. This can be observed from the constellation points represented by the red triangles in Fig. \ref{fig:Constellation-I_II}. Hence, the probability of error for $u_2$ with incorrect $u_1$ decoding, considering both $x_1^{(0)}$ and $x_1^{(1)}$ bit transmission scenarios, is (\ref{eq:P_e_u2_II}).

The conditional expressions in (\ref{eq:P_e_u2_II}) are obtained in the same way as in (\ref{eq:P_e_u2_I}) except that $x_1$ is decoded incorrectly in case II. By using conditional probability law, the probability of error for $u_2$ considering $x_1$ is decoded erroneously, represented using the Gaussian $\mathcal{Q}$ function, is given by

\begin{equation}\label{eq:P_e_u2_II_Q}
  P_e^{II}(u_2) = \frac{1}{2} \left[ \mathcal{Q}(C) +\mathcal{Q}(Z) -\mathcal{Q}(D)\right].
\end{equation}
where $Z$ is defined in (\ref{eq:Y_Z}), $C$ and $D$ are RVs, defined as

\begin{equation}\label{eq:C_D}
  \begin{split}
    C = \frac{2\Upsilon_1}{\sqrt{N_0/2}} + \frac{\Upsilon_2}{\sqrt{N_0/2}}, \\
    D = \frac{2\Upsilon_1}{\sqrt{N_0/2}} - \frac{\Upsilon_2}{\sqrt{N_0/2}}.
   \end{split}
 \end{equation}
Then, the total probability of error for $u_2$ can be found as the sum of both the cases given by (\ref{eq:P_e_u2_I_Q}) and (\ref{eq:P_e_u2_II_Q})

\begin{equation}\label{eq:P_e_u2_Q}
  \begin{split}
    P_e(u_2) &= P_e^I(u_2) + P_e^{II}(u_2), \\
    P_e(u_2) = \mathcal{Q}(\sqrt{\gamma_{\Upsilon_2}}) &+ \frac{1}{2}\left[-\mathcal{Q}(Y) + \mathcal{Q}(C) +\mathcal{Q}(Z) -\mathcal{Q}(D)\right].\nonumber
  \end{split}
\end{equation}

The RV $C$, i.e., sum of two i.n.i.d Nakagami-$m_i$ RVs, can be approximated by a Nakagami-$m_{R_2}$ RV with fading parameter $m_{R_2}$ and average power $\Omega_{R_2}$, with PDF given in (\ref{eq:PDF_nak}) as proved in Lemma 2, where $\Omega_{C_1}=4 \Omega_{1}$ and $\Omega_{C_2}=\Omega_{2}$ . The RV $D$, i.e., difference of two i.n.i.d Nakagami-$m_i$ RVs, can be approximated by a Gaussian (normal) RV, with mean $\mu_{W_2}$ and variance $\sigma^2_{W_2}$, with PDF given in (\ref{eq:normal_PDF}) as proved in Lemma 2, where $\Omega_{D_1}=4 \Omega_{1}$ and $\Omega_{D_2}=\Omega_{2}$ .

Therefore, the average probability of error for $u_2$ becomes

\begin{multline}\label{eq:Pu2_average}
  \overline{P_e(u_2) } = \Phi(m_2, \gamma_{\Upsilon_2}) +\frac{1}{2}\left[ -\Phi(m_{R_1},\Omega_{R_1}) +\Phi(m_{R_2},\Omega_{R_2})
  +  \Lambda(\mu_{W_1},\sigma^2_{W_1})-\Lambda(\mu_{W_2},\sigma^2_{W_2}) \right].
\end{multline}

\section{Numerical Results and Simulations}

\begin{figure*}[h]
  \centering
  \subfloat[$\Gamma_1=1, \Gamma_2=0.9$]{\includegraphics[width= 3.35in]{BER_m1_plot1.pdf}\label{fig:BER_plots_1}}
  \subfloat[$\Gamma_1=1, \Gamma_2=0.6$]{\includegraphics[width= 3.35in]{BER_m1_plot2.pdf}\label{fig:BER_plots_2}}\\
  \subfloat[$\Gamma_1=1, \Gamma_2=0.3$]{\includegraphics[width= 3.35in]{BER_m1_plot3.pdf}\label{fig:BER_plots_3}}
  \subfloat[$\Gamma_1=1, \Gamma_2=0.7, 0.2$]{\includegraphics[width= 3.35in]{BER_AWGN}\label{fig:BER_plots_4}}
  \caption{BER plots of BSN-1 and BSN-2 for fading ($m_1=4, \Omega_1 =1,m_2=1, \Omega_2 =0$) and fading-free scenarios.}
  \label{fig:BER_plots}
\end{figure*}

This section investigates the performance of NOMA enhanced bistatic BackCom system consisting of a single cluster of two-BSNs and presents the numerical results by evaluating the BER expressions derived in the previous section. The results are validated with Monte Carlo simulations and are found to match the derived expressions in this paper. The channel between BSNs and BSR is modeled as Nakagami-$m$ fading channel, and both BSN and BSR are assumed to be equipped with a single antenna. The transmitted symbols for both users are selected uniformly from a BPSK constellation. Unless stated otherwise, the figures are plotted for fading channel conditions given as $m_1=4, \Omega_1 =1,m_2=1, \Omega_2 =0.5.$

\subsection{Analysis Validation}

In Fig. \ref{fig:BER_plots}, numerical and simulated BER of the NOMA enhanced BackCom system is plotted against transmit SNR of both BSNs. A Nakagami-$m$ fading channel ($m_1=4, \Omega_1=1$) and a Rayleigh fading channel ($m_2=1, \Omega_2=1$) is taken in Fig. \ref{fig:BER_plots}\ref{sub@fig:BER_plots_1}, \ref{fig:BER_plots}\ref{sub@fig:BER_plots_2} and \ref{fig:BER_plots}\ref{sub@fig:BER_plots_3} for the fading scenario of BSN-1 and BSN-2, respectively. Three different pairs of reflection coefficient values (\(\Gamma_1\), \(\Gamma_2\)) are considered for the analysis. As can be seen from the figure, numerical results obtained using (\ref{eq:Pu1_avergae}) and (\ref{eq:Pu2_average}) match the simulation results for different pairs of \(\Gamma_1\) and \(\Gamma_2\) values thus validating the theoretical analysis. It can also be observed that a larger separation in reflection coefficient values results in better BER behavior for NOMA enhanced BackCom. This is because, by lowering \(\Gamma_2\), the IUI experienced by BSN-1 is decreased, resulting in better BER performance of BSN-1 due to the efficient utilization of the NOMA principle. As the BER performance of BSN-2 depends on the successful SIC operation of BSN-1, therefore, lowering the IUI indirectly influences the performance of BSN-2 as evident from its improved performance.

The BER plots are also given in Fig. \ref{fig:BER_plots}\ref{sub@fig:BER_plots_4} for a fading-free scenario by taking an arbitrarily large value of $m_1$ and $m_2$ to simulate a pure AWGN channel. The fading-free scenario provides a significant improvement in BER performance. Furthermore, similar behavior of larger separation resulting in better performance is also observed in fading-free scenarios.

\subsection{Effects of the reflection coefficients}

\begin{figure}[hb]
  \vspace{-.75em}
  \centering
  \includegraphics[width=.65\columnwidth]{Eff_bits_Plot_r2_vary_m1_opt.pdf}
  \caption{The normalized average of effectively decoded bits for BSN-1 and BSN-2 against $\Gamma_2$ while $\Gamma_1=0.7$.}
  \label{fig:Eff_bits_Plot_r2_vary}

\end{figure}

In Fig. \ref{fig:Eff_bits_Plot_r2_vary}, we investigate the effect of reflection coefficient on the normalized average of total effectively decoded bits of $u_1$ and $u_2$. The total effectively decoded bits correspond to the non-erroneous transmission of BSNs' bits over the total number of bits transmitted by BSN-1 and BSN-2. It is assumed that the reflection coefficient for the BSN-1, $\Gamma_1$, is set as 0.7 for the analysis. The results are plotted for three values of transmit SNRs, i.e., 0, 10, and 20 dB.

From Fig. \ref{fig:Eff_bits_Plot_r2_vary}, we can observe that the total normalized effective bits of $u_1$ and $u_2$ decrease with an increase in $\Gamma_2$ value. Therefore, the system performance is improved by setting a low value of $\Gamma_2$. This is again because, by setting a low value of $\Gamma_2$, the interference experienced by BSN-1 is reduced, thus, its BER performance is improved resulting in the greater non-erroneous transmission of bits by the system. However, it is to be noted from figure, that $\Gamma_2$ has a minimum value below which the performance of the system starts to degrade as BSN-2 is not able to decode itself at such a small value resulting in higher transmission errors.

\begin{figure}[t!]
  \centering
  \subfloat[Fading with transmit SNR = 15 dB]{\includegraphics[width= 3.24in]{Contour_m1_plot1.pdf}\label{fig:Contour_plots_1}}\\
  \subfloat[Fading with transmit SNR = 20 dB]{\includegraphics[width= 3.24in]{Contour_m1_plot2.pdf}\label{fig:Contour_plots_2}}\\
  \subfloat[Fading-free with transmit SNR = 15 dB]{\includegraphics[width= 3.24in]{Contour_m1_plot3.pdf}\label{fig:Contour_plots_3}}
  \caption{BER contour plot of BSN-1 and BSN-2 by varying $\Gamma_1$ and $\Gamma_2$ values for fading and fading-free scenarios}
  \label{fig:Contour_plots}

\end{figure}

The $\Gamma_2$ values are also marked for each transmit SNR which provides the best system performance in terms of the effective bits transmitted. For transmit SNRs of 0, 10, and 20 dBs, the optimal $\Gamma_2$ is 0.67, 0.27 and 0.21, respectively. It is evident from the fact that by lowering the transmit SNR, a greater $\Gamma_2$ value is needed by BSN-2 to decode itself successfully. We also find that the maximum normalized total effective bits turns out to be 0.9791 for the 20 dB SNR scenario.

In Fig. \ref{fig:Contour_plots}, a contour plot of BER is plotted by varying the reflection coefficient pair of BSN-1 and BSN-2. The reflection coefficients of both BSNs are varied from 0.01 to a maximum value of 1 by always keeping \(\Gamma_1 > \Gamma_2\) as found from the previous result in Fig. \ref{fig:BER_plots}. The BER of BSN-1 is represented by a dotted line while the BER of BSN-2 is represented by a solid line. It can be observed from the contour plot that for any specific value of \(\Gamma_1\), there exists a range of \(\Gamma_2\) values smaller than \(\Gamma_1\) for which we can achieve acceptable performance in a NOMA-BackCom system. For the fading case (see Fig. \ref{fig:Contour_plots}\ref{sub@fig:Contour_plots_1} and \ref{fig:Contour_plots}\ref{sub@fig:Contour_plots_2}), it can be observed that very little improvement is possible in BER unless either the transmit SNR is increased by placing the BSNs closer to CE or fading channel condition is boosted by removing obstructions in line-of-sight (LoS) path. The contour plots of the fading-free case is plotted in Fig. \ref{fig:Contour_plots}\ref{sub@fig:Contour_plots_3} for improvement comparison due to less channel severity.

\subsection{Effect of the Nakagami-$m$ fading parameter}

Now, we investigate the impact of the fading parameter $m$ on the BER performance of each BSN for a NOMA-BackCom system with two different sets of reflection coefficients. The BER plots are shown in Fig. \ref{fig:m_vary} for transmit SNRs of $u_1$ and $u_2$ as 20 and 15 dBs, respectively. As can be observed from figure, the BER performance of both users depend strongly on the fading parameters of the channel. It can be seen that for suitable reflection coefficient pairs, the fading parameters affects the performance of BSN-1 more than BSN-2.

\begin{figure}[h]
  \centering
  \includegraphics[width=.7\columnwidth]{m_vary.pdf}
  \caption{BER plots of BSN-1 and BSN-2 for various $m$ values.}
  \label{fig:m_vary}

\end{figure}

\begin{figure}[t!]

  \centering
  \subfloat[Comparison of total effective bits]{\includegraphics[width= 4.2in]{Effec_bits_total_single.pdf}\label{fig:OMA-comparison_1}}

  \\
  \subfloat[Comparison of individual effective bits]{\includegraphics[width= 4.2in]{Effec_bits_Good_single.pdf}\label{fig:OMA-comparison_2}}
  \caption{Comparison of normalized effective bits transmitted of a NOMA-BackCom system ($\Gamma_1=0.7, \Gamma_2=0.2$) against OMA-TDMA scheme.}
  \label{fig:OMA-comparison}
  \vspace{-.75em}
\end{figure}

\subsection{Comparison with OMA-TDMA scheme}
In this section, we compare the performance of the NOMA-BackCom system with an OMA-TDMA transmission scheme in a BackCom system. Fig. \ref{fig:OMA-comparison} illustrates the increase in effective non-erroneous bits transmission by employing the NOMA scheme in a BackCom system of a single cluster with two BSNs as compared to an OMA-TDMA transmission scheme with two BSNs. It can be observed that the NOMA scheme indeed outperforms the OMA scheme due to the simultaneous transmission of two bits to the reader in a single time slot even though the OMA-TDMA scheme experiences no IUI from the second BSN. Individually, the TDMA BSN always has better BER performance as compared to NOMA BSN, i.e., a TDMA user has a greater number of successful transmissions as shown in Fig. \ref{fig:OMA-comparison} \ref{sub@fig:OMA-comparison_2}. However, the combined performance of NOMA scheme is better as the time spent by BSN on each time slot is doubled under NOMA scheme as compared to TDMA scheme. At high SNR though, the BER difference between an individual NOMA and TDMA user becomes negligible.

\section{Conclusion}
This work has presented the design and analysis of a NOMA enhanced bistatic BackCom system for a battery-less smart communication paradigm employed in an IoT scenario. We have derived the closed-form BER expressions for a cluster of two BSNs with imperfect SIC under Nakagami-$m$ fading channel. Furthermore, the PDFs of the sum and difference of two i.n.i.d Nakagami-$m$ distributions are also accurately approximated. All the derived expressions are verified with the simulations under different scenarios. Based on these expressions, we have evaluated the performance of the system in terms of the  reflection coefficients. We have also found that the increment of SNR with unsuitable reflection coefficients does not lead to a better system performance, hence highlighting the significance of setting proper reflection coefficients according to the scenario. This necessitates an optimization study as a future work where reflection coefficients can be optimized such that the system performance in terms of BER or effective transmitted bits can be improved. Further future extensions include BER analysis for higher modulation schemes with a higher number of BSNs in different fading environments.

\appendices
\section{K-S Test for the sum of two i.n.i.d  Nakagami-$m_i$ RVs}
The random number generation routine is repeated $N$ times to collect samples of the RV $Y$, i.e., $\{y_1,y_2,...,y_N \}$, with empirical cumulative distribution function (CDF) $\widehat{F}_Y $. The hypothesized CDF is that of Nakagami-$m_{R_1}$ distribution, $F_{R_1}$. The statistic used for goodness-of-fit known as K-S statistic is the maximum difference between the empirical CDF and hypothesized CDF, given by \cite{Press2002}

\begin{equation}\label{eq:KS_statistic}
  \widehat{D}_f =  \underset{{y}}{\operatorname{sup}}|\widehat{F}_Y(y_i)-F_{R_1}(y_i)|.
\end{equation}
The critical value is found to be $\widehat{c} = 0.0192$ for $N = 5000$ samples against the level of significance $\widehat{a} = 0.05$. The null hypothesis for testing is given as

\begin{equation}\label{eq:Null_hypothesis}
  H_0 : F_Y = F_{R_1}.
\end{equation}
The null hypothesis is accepted if $\widehat{D}_f \leq \widehat{c}$, i.e., $F_Y=F_{R_1}$ and rejected otherwise. The K-S test is conducted for three set of parameters, i.e., \{$m_1=1, \Omega_1 =1,m_2=1, \Omega_2 =1$\}, \{$m_1=3, \Omega_1 =1,m_2=1, \Omega_2 =1$\} and \{$m_1=3, \Omega_1 =1,m_2=3, \Omega_2 =0.5$\} and K-S statistic, $\widehat{D}_f $, calculated using (\ref{eq:KS_statistic}) are

\begin{equation}
  \begin{split}
    \widehat{D}_f &= 0.0122 \text{ for } m_1=1, \Omega_1 =1,m_2=1, \Omega_2 =1,\\
    \widehat{D}_f &= 0.0082 \text{ for } m_1=2, \Omega_1 =1,m_2=1, \Omega_2 =1,\\
    \widehat{D}_f &= 0.0077 \text{ for } m_1=3, \Omega_1 =1,m_2=3, \Omega_2 =0.5.\\
  \end{split}
\end{equation}
It can be observed, that $\widehat{D}_f \leq \widehat{c}$ for each parameter set and $H_0$ cannot be rejected. Hence, Nakagami-$m$ distribution closely matches the distribution of the sum of two Nakagami-$m$ RVs and can be used to approximate RV $Y$.

\section{K-S Test for the difference of two i.n.i.d  Nakagami-$m_i$ RVs}
Again, the Random number generation routine is repeated $N$ times to collect samples of the RV $Z$ , i.e., $\{z_1,z_2,...,z_N \}$, with empirical cumulative distribution function (CDF) $\widehat{F}_Z $. The hypothesized CDF is that of normal distribution, $F_{W_1}$.

The critical value is found to be $\widehat{c} = 0.0192$ for $N = 5000$ samples against the level of significance $\widehat{a} = 0.05$. The null hypothesis for testing is given as

\begin{equation}\label{eq:Null_hypothesis2}
  H_0 : F_Z = F_{W_1}.
\end{equation}

The K-S test is conducted for three set of parameters, i.e., \{$m_1=1, \Omega_1 =1,m_2=1, \Omega_2 =1$\}, \{$m_1=3, \Omega_1 =1,m_2=1, \Omega_2 =1$\} and \{$m_1=3, \Omega_1 =1,m_2=3, \Omega_2 =0.5$\} and K-S statistic, $\widehat{D}_f $, calculated using (\ref{eq:KS_statistic}) for each set, given as

\begin{equation}
  \begin{split}
    \widehat{D}_f &= 0.0109 \text{ for } m_1=1, \Omega_1 =1,m_2=1, \Omega_2 =1,\\
    \widehat{D}_f &= 0.0141 \text{ for } m_1=2, \Omega_1 =1,m_2=1, \Omega_2 =1,\\
    \widehat{D}_f &= 0.0140 \text{ for } m_1=3, \Omega_1 =1,m_2=3, \Omega_2 =0.5.\\
  \end{split}
\end{equation}

$H_0$ cannot be rejected for each parameter set as $\widehat{D}_f \leq \widehat{c}$. Hence, normal distribution closely matches the distribution of the difference of two Nakagami-$m$ RV and can be used to approximate RV $Z$.

\section{Proof of Lemma 2}
Let $W$ be a normal distributed RV with parameters mean $\mu$ and variance $\sigma^2$, i.e., $W\sim N(\mu,\sigma^2)$ and $Z$ be a standard normal distributed RV, i.e., $Z\sim N(0,1)$. Both being independent, then $\mathcal{Q}(w)=\mathbb{P}(Z>w)$ as $\mathcal{Q}(x)$ is the probability that a standard normal RV takes a value greater than $x$. This is similar to $\Lambda$ because the integrand in $\Lambda$ is the product of $\mathbb{P}(Z>w) =\mathbb{P}(Z>W|W=w)$ and the marginal density of normal RV, $W$, i.e., $f_W(w)$, which comes out to be $\mathbb{P}(Z>W)$. As we know that distribution of the difference of two normal RVs is again a normal RV with subtracted means but added variances, i.e., $Z-W \sim N(-\mu,\sigma^2+1)$, therefore

\begin{equation}
  \Lambda = \mathbb{P}(Z>W) = \mathbb{P}(Z-W>0) = \mathcal{Q}\bigg(\frac{\mu}{\sigma^2+1}\bigg).
\end{equation}

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{IEEEtran}
\bibliography{IEEEabrv,TII_References}

\end{document}

config: arxiv


TITLE: Ad-invariant scalar product on complex vector space
QUESTION [0 upvotes]: Let's say we have a Lie algebra $\mathfrak{g}$ over a complex vector space, $z_1,z_2,z_3 \in \mathfrak{g}$. What is the defining property for an ad-invariant real valued scalar product
\begin{equation}
\langle z_1,[z_2,z_3] \rangle = \langle [z_1,z_2^*],z_3 \rangle
\end{equation}
or
\begin{equation}
\langle z_1,[z_2,z_3] \rangle = \langle  [z_1,z_2],z_3 \rangle
\end{equation}

REPLY [2 votes]: First of all, the "correct" definition of invariance for a bilinear or hermitian form $\langle \cdot, \cdot\rangle$ on a vector space $V$ with respect to a Lie algebra representation $\rho: \mathfrak g \rightarrow \mathfrak{gl}(V)$ is:
$$\langle \rho(x).v,w\rangle = -\langle v, \rho(x).w\rangle \qquad\text{ for all } x \in \mathfrak g, v,w \in V. \quad (*)$$
For $\rho=ad$ (and e.g. the Killing form) this is often written in the way $\langle [a,b],c\rangle = \langle a, [b,c]\rangle$ you allude to, which in that case is equivalent to the above via $x=-b, a=v, c=w$ and using $[r,s]=-[s,r]$. Still, the above should be preferred due to its much greater generality.
But also with that better definition, you will notice (analogously to your own comment), you run into a problem if $V$ is complex, $\langle \cdot, \cdot \rangle$ is hermitian (i.e. conjugate-linear in, say, the second components), and if you are allowed to multiply $\rho(x)$ through with complex scalars. Indeed, $(*)$ implies that for all scalars $a$ that you can multiply $x$ with and which commute with $\rho$:
$a\langle \rho(x).v,w\rangle = \langle a\rho(x).v,w\rangle = \langle \rho(ax).v,w\rangle \stackrel{(*)}= -\langle v, \rho(ax).w\rangle = -\langle v, a\rho(x).w\rangle =-\color{red}{\bar a}\langle v, \rho(x).w\rangle$
which divided by the original $(*)$ implies $a=\bar a$.
Now the way out is not to not allow complex $V$ or try to force the hermitian inner product to be "real valued" as in the comments, because that would force it to be identically $0$ (and hence not really a product).
Further, as regards your proposed first option, as several comments point out, there is no natural definition for something like $z^*$ in this setting if $\mathfrak g$ is a complex Lie algebra. Indeed, to define that, you would need to choose a real structure of that $\mathfrak g$. The proposed matrix trick also doesn't work in general: What if the conjugate transpose is not in the Lie algebra? (E.g. just take as $\mathfrak g$ some Lie algebra of upper triangular matrices.)
The actual way out is to realize (pun intended) that all this only works for real Lie algebras $\mathfrak g$, so that in the above, $a \in \mathbb R$. Actually, it should turn out later in the theory that such invariant inner products need a kind of compactness which only works for certain real Lie algebras, not complex ones. This way, the theory is treated in Bourbaki's Lie Groups and Algebras, chapter IX (Compact Lie Groups), which starts with a discussion of exactly this kind of invariant hermitian products.


TITLE: Inverse 4x4 Matrix using Calculator
QUESTION [1 upvotes]: My calculator only supports a 3x3, 3x2, 3x1, 2x3, 2x2, 2x1, 1x3, 1x2, 1x1 matrices. Is there any chance I can get the inverse of the 4x4 using my calculator using the matrix mode?

REPLY [0 votes]: In principle, you could partition your matrix into $2 \times 2$ blocks (i.e. a $2 \times 2$ matrix whose elements are $2 \times 2$ matrices) and then apply "Block Inversion" :  have a look at this article.
But unfortunately the resulting formula is somewhat complicated. However it is instructive to know about the division of a matrix into blocks.


text:
\begin{document}
\title{Lusin's Theorem and Bochner Integration}
\author{Peter A. Loeb\\Department of Mathematics, University of
Illinois\\1409 West Green Street,
Urbana, Illinois 61801, U.S.A.\\e-mail: loeb@math.uiuc.edu
\and Erik Talvila\thanks{The work of both authors here was supported in part by the NSERC and the University of Alberta. The authors thank David Ross for a helpful pointer to the literature in \cite{fremlin0} and \cite{kupkaprikry}.}\\Department of Mathematics and Statistics \\ University College of the Fraser Valley\\Abbotsford, BC, Canada V2S 7M8\\e-mail: Erik.Talvila@ucfv.ca}
\date{To appear in {\it  Scientiae Mathematicae Japonicae.}
Accepted January 13, 2004.}
\subjclass{Primary 28A20, 28B05; Secondary 26A39}
\keywords{Bochner integral, Lebesgue point, Lusin's theorem, Riemann sum, gauge function.}

\maketitle

\begin{abstract}
It is shown that the approximating functions used to define the Bochner
integral can be formed using geometrically nice sets, such as balls, from a
differentiation basis. Moreover, every appropriate sum of this form will be
within a preassigned $\varepsilon$ of the integral, with the sum for the local
errors also less than $\varepsilon$. All of this follows from the ubiquity of
Lebesgue points, which is a consequence of Lusin's theorem, for which a simple
proof is included in the discussion.
\end{abstract}

\section{Introduction}

An attractive feature of the Henstock--Kurzweil integral for the real line is
that it can be defined in terms of Riemann sums using intervals. In
\cite{loebtalvila}, we showed that for more general spaces, Lebesgue points
and points of approximate continuity can be used to approximate a Lebesgue
integral $\int_{X}f~d\mu$ with sums employing disjoint, geometrically nice
sets $S_{i}$ covering all but a set of measure $0$ of $X$. We also showed that
any such sum will be within a preassigned $\varepsilon$ of the integral,
provided each term $f(x_{i})\mu(S_{i})$ of the sum has the property that
$S_{i}$ is contained in a ball about $x_{i}$ of radius at most $\delta(x_{i})$,
where $\delta$ is a \textquotedblleft gauge\textquotedblright\ function
determined by $f$ and $\varepsilon$.

In this article, we show that a similar result holds for the Bochner integral,
where the domain of the integrand is a measure space
$\left(  X,\mathcal{M},\mu\right)$ on which is defined a differentiation basis.
Although somewhat more general settings are possible, we will assume that $X$ is a finite
dimensional normed vector space and the differentiation basis is obtained
using the Besicovitch or Morse covering theorem (see Section \ref{Covering}).
The geometrically nice sets we will use for the approximating functions will
be balls or the more general starlike sets described in Section~\ref{Covering}.
We will assume that $\mu$ is a complete \textbf{Radon measure}; i.e., $\mu$
is a regular measure on $\mathcal{M}$, which includes the Borel sets, and
compact sets have finite measure. By a \textbf{measurable} set, we will always
mean a set in $\mathcal{M}$. We will let $\mathbb{N}$ denote the natural
numbers and $\mathbb{R}$ the real numbers.

In what follows, an integrand $f$ will take its values in a Banach space
$\left(  Y,\left\Vert \cdot\right\Vert \right)  $ and will be
$\mu$\textbf{-measurable}, meaning, there is a sequence of simple functions
$f_{n}$ with $\lim_{n}\left\Vert f_{n}-f\right\Vert =0$ $\mu$-a.e. on $X$.
This will allow us to use Lusin's theorem. An elementary proof of that
theorem, simple even for real-valued functions on $\mathbb{R}$, is given in
the next section.

Our approximating sums will be constructed using a gauge function $\delta$
mapping $X$ into $(0,1]$. In the theory of Henstock--Kurzweil and McShane
integration, the appearance of a gauge function is somewhat mysterious. We
show in proving Theorem \ref{integration} how the properties of Lebesgue
points (discussed below in Section \ref{Covering}) can be used to determine an
appropriate gauge.

We note that the Bochner integral has been studied in terms of Riemann sums
over finite partitions of a metric space in \cite{Neerven} and over
generalized McShane partitions of a measure space in \cite{fremlin} and
\cite{dipiazza}. See also \cite{dipiazzamusial}. None of these papers,
however, works with approximations using geometrically nice sets.

\section{Lusin's Theorem}

\label{LusinSec}

Recall that a function $f$ from $X$ into a topological space $(Y,\mathcal{T})$
is \textbf{Borel measurable} if the inverse image of each open set in $Y$ is
in $\mathcal{M}$. In our case, where $Y$ is a Banach space and $f$ is $\mu
$-measurable, it follows from Theorem III.6.10 of \cite{dunfordschwartz} that
$f$ is Borel measurable on $X$. We note that it follows from the same theorem
and a deep result of D. H. Fremlin (Theorem 2B in \cite{fremlin0} or Theorem
4.1 in the expository article \cite{kupkaprikry} by J. Kupka and K. Prikry)
that if $Y$ is a metric space and  $f:X\rightarrow Y$ is Borel measurable,
then $f$ is $\mu$-measurable.

In any case, Lusin's theorem holds for the restriction of $f$ to a set
$\Omega\subseteq X$ with $\mu\left(  \Omega\right)  <+\infty$, and is easily
extended to all of $X$ using the $\sigma$-finiteness of $\mu$. Here is the
general statement of the theorem for the case of a finite measure space, with
an elementary proof that is appropriate even for the simplest setting.

\begin{theorem}
[Lusin]\label{Lusin}Let $Y$ be a topological space with a countable base
$\left\langle V_{n}\right\rangle $ for the topology, and let $f$ be a Borel
measurable function from a Radon measure space of finite measure
$(\Omega,\mathcal{A},\mu)$ into $Y$. Given $\varepsilon>0$, there is a compact
set $K$ with $\mu(\Omega\setminus K)<\varepsilon$ such that $f$ restricted to
$K$ is continuous.
\end{theorem}

\noindent \textbf{Proof:} Fix compact sets $K_{n}\subseteq f^{-1}[V_{n}]$ and $K_{n}^{\prime}
\subseteq\Omega\setminus f^{-1}[V_{n}]$ for each $n$ so that $\mu
(\Omega\setminus K)<\varepsilon$ when $K:=\bigcap\nolimits_{n}(K_{n}\cup
K_{n}^{^{\prime}})$. Given $x\in K$ and an $n$ with $f(x)\in V_{n}$,\ \ $x\in
O:=\Omega\setminus K_{n}^{\prime}$ and $f\left[  O\cap K\right]  \subseteq
V_{n}$. $\square$

\begin{remark}
\begin{rm}
In Oxtoby's text \cite{oxtoby} a similar principle is used in a more complex
proof to show that a measurable $f:\mathbb{R}\rightarrow\mathbb{R}$ is
continuous when restricted to a large measurable subset of $\mathbb{R}$.
\end{rm}
\end{remark}

\section{Covering Theorems}

\label{Covering}

Our integration result is based on calculations using a covering theorem. Here
we use either Besicovitch's theorem \cite{besicovitch} (also
see\ \cite{furediloeb}) for a covering by balls, or the theorem of Morse
\cite{morse} involving more general sets. In \cite{loebtalvila}, we have
established strengthened versions of these theorems that hold for our finite
dimensional normed vector space $X$. Moreover, these covering theorems are
also valid for a space that is locally isometric to $X$ since one only needs
to bound the cardinality of a finite collection of sets with small diameter
forming a \textquotedblleft$\tau$-satellite configuration\textquotedblright
as defined in \cite{furediloeb} and \cite{loebtalvila}. We leave it to the
interested reader to consider our results in this more general setting, as
well as settings using Vitali's covering theorem.

We will denote the closed (and compact) ball in $X$ with center $a$ and radius
$r>0$ by $B(a,r):=\{x\in X:\Vert x-a\Vert\leq r\}$. The sets used by Morse
involve a parameter $\lambda\geq1$; they are closed balls when $\lambda=1$.

Given $\lambda\geq1$ and $a\in X$, we say that a set $S(a)\subseteq X$ is a
\textbf{Morse set} or $\lambda$\textbf{-Morse set} associated with $a$ and
$\lambda$ if it satisfies two conditions. First, $S(a)$ must be
$\lambda$\textbf{-regular}. This means that there is an $r>0$ such that
$B(a,r)\subseteq S(a)\subseteq B(a,\lambda r)$. Second, $S(a)$ must be
\textbf{starlike} with respect to $B(a,r)$. This means that for each
$y \in B(a,r)$ and each $x \in S(a)$, the line segment
$\alpha y+(1-\alpha)x$,\ \ $0\leq\alpha\leq1$, is contained in $S(a)$.
We call $a$ the \textbf{tag} for $S(a)$. Note that the closure
$\operatorname*{cl}\left(  S(a)\right)$ of a
$\lambda$-Morse set $S(a)$ is again a $\lambda$-Morse set. We say that a
$\lambda$-Morse set $S(a)$ is $\delta$\textbf{-fine} with respect to a gauge
function $\delta$ defined at $a$ if $S(a)\subseteq B(a,\delta(a))$.

Suppose we are given $\lambda\geq1$, a Radon measure $\mu$ on $X$, and an open
subset $\Omega$ of $X$. We will call a collection $\mathcal{S}$ of
$\lambda$-Morse sets a \textbf{fine},\textbf{ measurable},
$\lambda$\textbf{-Morse cover }of $\Omega$ provided each set in
$\mathcal{S}$ is a measurable subset
of $\Omega$ and each $a\in\Omega$ is the tag of sets in $\mathcal{S}$ with
arbitrarily small diameters. A sequence $\left\langle S_{i}\right\rangle $
from such an $\mathcal{S}$ is said to be $\mu$\textbf{-exhausting of} $\Omega$
if it is a finite or countably infinite sequence that is pairwise disjoint,
and covers all but a set of $\mu$-measure $0$ of $\Omega$. A collection
$\mathcal{S}$ is called a $\mu$\textbf{-a.e.\hspace{0.03in}}$\lambda
$\textbf{-Morse} \textbf{cover} of $\Omega$ if it is a fine, measurable,
$\lambda$-Morse cover of $\Omega$ and for each nonempty open subset
$U\subseteq\Omega$ the collection
$\mathcal{S}_{U}=\left\{  S\in \mathcal{S}:S\subseteq U\right\}  $
has the property that for any gauge
function $\delta:U\rightarrow(0,1]$, there is a sequence of $\delta$-fine sets
in $\mathcal{S}_{U}$ that is $\mu$-exhausting of $U$. In \cite{loebtalvila} we
have established the following result for $\mu$ and $\Omega$.

\begin{proposition}
\label{prop1} A fine, measurable, $\lambda$-Morse cover $\mathcal{S}$ \ of
$\Omega$ is a $\mu$-a.e. $\lambda$-Morse\hspace{0.02in}cover of $\Omega$ if it
consists of closed sets or if for each set
$S\in\mathcal{S}$, $\mu(\Omega \cap(\operatorname*{cl}(S)\setminus S))=0$.
\end{proposition}

We have also shown that the conditions in Proposition~\ref{prop1} are
fulfilled by any measurable Morse cover $\mathcal{S}$ that is \textbf{scaled}.
This means that for each $S(a)\in\mathcal{S}$ and each $p\in(0,1]$, the set
$S^{(p)}(a)$ is also in $\mathcal{S}$ where $S^{(p)}(a)=\{a+px:a+x\in S(a)\}$.

\section{Approximate Continuity and Lebesgue Points}

\label{LebPt}

For this section, we fix a $\mu$-a.e. $\lambda$-Morse cover $\mathcal{S}$ of
an open set $\Omega\subseteq X$. We work with a $\mu$-measurable
$f:\Omega\rightarrow Y$. The following notions depend on the choice of
$\mathcal{S}$.

\begin{definition}
A point $a\in\Omega$ is a \textbf{point of approximate continuity} for $f$ if
for all positive $\varepsilon$ and $\eta$ there is an $R>0$ such that if
$S(a)$ is a set in $\mathcal{S}$ with tag $a$ and $S(a)\subseteq B(a,R)$, then
for $E(a,\eta):=\{x\in S(a):\left\Vert f(a)-f(x)\right\Vert >\eta\}$ we have
$\mu(E(a,\eta))\leq\varepsilon\,\mu(S(a))$. A point $a\in\Omega$ is a
\textbf{Lebesgue point} of $f$ with respect to $f(a)$ if for any
$\varepsilon>0$ there is an $R>0$ such that if $S(a)$ is a set in
$\mathcal{S}$ with tag $a$ and $S(a)\subseteq B(a,R)$, then
\[
\int\nolimits_{S(a)}\left\Vert ~\!\!f(x)-f(a)\right\Vert \,\mu(dx)\leq
\varepsilon\,\mu(S(a)).
\]

\end{definition}

It is easy to see that if $a\in\Omega$ is a Lebesgue point of $f$ with respect
to $f(a)$, then $a$ is a point of approximate continuity for $f$. It follows
from the fact that $\mathcal{S}$ is a differentiation basis that if $g$ is a
$\mu$-integrable, nonnegative, real-valued function on $\Omega$, then $\mu
$-almost all points of $\Omega$ are Lebesgue points. (See, for example,
\cite{bliedtnerloeb}.) Moreover, if $A$ is a measurable subset of $\Omega$,
then almost all points of $A$ are \textbf{points of density}, that is, points
of approximate continuity with respect to the characteristic function
$\chi_{A}$ of $A$. Therefore, we have the following consequence of Lusin's theorem.

\begin{proposition}
If $f:\Omega\rightarrow Y$ is $\mu$-measurable, then $\mu$-almost all points
of $\Omega$ are points of approximate continuity for $f$.
\end{proposition}

\noindent \textbf{Proof:} If $\mu\left(  \Omega\right)  <+\infty$, then by Lusin's theorem
(Theorem \ref{Lusin}), there is an increasing sequence of compact sets
$K_{n}\subseteq\Omega$ such that for each $n$, $f|K_{n}$ is continuous and
$\mu\left(  \Omega\backslash\cup_{n}K_{n}\right)  =0$. For this case, the
result follows from the fact that for each $n$, $\mu$-almost every point of
$K_{n}$ is a point of density of $K_{n}$. The general case follows since
$\Omega$ has $\sigma$-finite measure. $\square$

\bigskip

We also have the following relationship between points of approximate
continuity and Lebesgue points.

\begin{proposition}
\label{lebisleb}If $f:\Omega\rightarrow Y$ is $\mu$-measurable and $a\in
\Omega$ is a point of approximate continuity for $f$ and also a Lebesgue point
for $\left\Vert f\right\Vert $ with respect to $\left\Vert f(a)\right\Vert $,
then $a$ is a Lebesgue point for $f$ with respect to $f(a)$.
\end{proposition}

\noindent \textbf{Proof:} Fix $\varepsilon>0$. Let $c=\left\Vert f(a)\right\Vert $
and choose $R>0$ so that if $S$ is a set in $\mathcal{S}$ with tag $a$ and
$S\subseteq B(a,R)$, then
\[
\int_{S}\left\vert ~\left\Vert f(x)\right\Vert -c~\right\vert \ \mu
(dx)<\varepsilon\cdot\mu\left(  S\right)  ,
\]
and for $E:=\{x\in S:\left\Vert f(x)-f(a)\right\Vert >\varepsilon\}$, we have
$\mu(E)\leq\frac{\varepsilon}{2c+1}\cdot\mu\left(  S\right)  $. Now
\[
\int_{E\cap\left\{  \left\Vert f\right\Vert >c\right\}  }\left(  \left\Vert
f(x)\right\Vert -c\right)  \ \mu(dx)<\varepsilon\cdot\mu\left(  S\right)  ,
\]
whence
\[
\int_{E\cap\left\{  \left\Vert f\right\Vert >c\right\}  }\left\Vert
f(x)\right\Vert \ \mu(dx)\leq2\varepsilon\cdot\mu\left(  S\right)  .
\]
Therefore,
\begin{align*}
&  \int_{S}\left\Vert f(x)-f(a)\right\Vert \ \mu(dx)\\
&  \leq\int_{S\backslash E}\varepsilon\ \mu(dx)+\int_{E}2c\ \mu(dx)+\int
_{E\cap\left\{  \left\Vert f\right\Vert >c\right\}  }\left\Vert
f(x)\right\Vert \ \mu(dx)\leq4\varepsilon\cdot\mu\left(  S\right) .
\end{align*}
Since the choice of $\varepsilon>0$ is arbitrary, the result follows.
$\square$

\bigskip

\begin{remark}
\begin{rm}
Proposition \ref{lebisleb} also holds for the $\mu$-null set consisting of
points $a\in\Omega$ that are Lebesgue points of $\left\Vert f\right\Vert $
with respect to values different from $\left\Vert f(a)\right\Vert $.
\end{rm}
\end{remark}

\section{Integration}

\label{Integration}

Recall that if $\mu\left(  \Omega\right)  <+\infty$, then a $\mu$-measurable
function $f:\Omega\rightarrow Y$ is \textbf{Bochner integrable} (with respect
to $\mu$) if there is a sequence of simple functions
$f_{n}:$ $\Omega \rightarrow Y$ with
$\lim_{n}\int_{\Omega}\left\Vert f_{n}-f\right\Vert d\mu=0$.
In this case, the sequence of integrals $\int_{\Omega}f_{n}~d\mu$ is
Cauchy in $Y$, and the limit is the Bochner integral of $f$. Also, $f$ \ is
Bochner integrable if and only if the norm $\left\Vert f\right\Vert $ is
integrable. (See, for example, Chapter II of \cite{diesteluhl}.) We will
consider the analogous approximations and integral for the case that
$\left\Vert f\right\Vert $ is integrable on our $\sigma$-finite measure space
$\left(  X,\mathcal{M},\mu\right)  $, with the understanding that a simple
function is defined on all of $X$ but must vanish off of a set of finite
measure. In Theorem 12 of \cite{loebtalvila}, we have established a necessary
and sufficient condition for the integrability of a real-valued function, such
as $\left\Vert f\right\Vert $, in terms of approximations by Riemann sums
using sets from a $\mu$-a.e. Morse cover. Now, assuming that $\left\Vert
f\right\Vert $ is $\mu$-integrable on $X$, we show that the Bochner integral
of $f$ can also be approximated by the integral of any appropriate simple
function using Morse sets.

\begin{theorem}
\label{integration}Assume $\left\Vert f\right\Vert $ is $\mu$-integrable on
$X$, and fix $\varepsilon>0$. Given $\lambda\geq1$, and any $\mu$-a.e.,
$\lambda$-Morse cover $\mathcal{S}$ \ of $\ X$, there is a gauge function
$\delta\!:\!X\rightarrow(0,1]$ such that for any finite or countably infinite
sequence $\left\langle S_{i}(x_{i})\right\rangle $ of $\delta$-fine sets from
$\mathcal{S}$ that is $\mu$-exhausting of $X$, the function
$\sum_{i} f(x_{i})\chi_{S_{i}}$ approximates $f$ in the sense that
\[
\int_{X}\left\Vert f(y)-\sum\nolimits_{i}\,f(x_{i})\chi_{S_{i}}
\,(y)\right\Vert ~\mu\left(  dy\right)  <\varepsilon
\]
and the absolute sum of local errors is small; that is,
\begin{equation}
\sum\nolimits_{i}\left\Vert \int_{S_{i}}f(y)~\mu(dy)-f(x_{i})\mu
(S_{i})\right\Vert <\varepsilon. \label{localerrorsum}
\end{equation}
The same approximation holds for the restriction of $f$ \ to a large open ball
$\Omega$ about the origin with
$\int_{X\diagdown\Omega}\left\Vert f\right\Vert <\varepsilon$,
but then there is an $m\in\mathbb{N}$ such that for any $n\geq m$,
\begin{align*}
\lefteqn{\left\Vert \int_{X}f(y)~\mu\left(  dy\right)  -\sum\nolimits_{i=1}
^{n}\,f(x_{i})~\mu\left(  S_{i}\right)  \right\Vert }\\
&  \leq\int_{X}\left\Vert f(y)-\sum\nolimits_{i=1}^{n}\,f(x_{i})\chi_{S_{i}
}\,(y)\right\Vert ~\mu\left(  dy\right)  <3\varepsilon.
\end{align*}

\end{theorem}

\noindent \textbf{Proof:} Fix $\gamma>0$ so that for each $E\subseteq X$ with
$\mu(E)<\gamma $,\ $\int_{E}\left\Vert f\right\Vert ~d\mu<\varepsilon/4$.
Let $L$ be the set of points $x\in X$ that are Lebesgue points of $f$ with
respect to $f(x)$, and let $A:=X\setminus L$. Since $\mu\left(  A\right)  =0$,
there is an open set $G$ containing $A$ with $\mu(G)<\gamma$.
For each $n\in\mathbb{N}$, we set
$A_{n}=\{x\in A:n-1\leq\left\Vert f(x)\right\Vert <n\}$. The sets $A_{n}$ are
disjoint and $\mu$-null with union $A$. For each $n\in\mathbb{N}$, fix an open
set $G_{n}$ with $G\supseteq$ $G_{n}\supseteq A_{n}$ and
$\mu(G_{n})<\varepsilon/\left(  n\cdot2^{n+2}\right)  $. For each $x\in A_{n}$, we
choose $\delta(x)<1\wedge\sup\{s:B(x,s)\subseteq G_{n}\}$. Then for any finite
or countably infinite disjoint sequence of $\ \delta$-fine sets $S_{i}$ from
$\mathcal{S}$ with tags $\ x_{i}$ \ in $\ A$, we have
\[
\sum\limits_{i}\left\Vert f(x_{i})\right\Vert \,\mu(S_{i})\leq\sum
\limits_{n=1}^{\infty}\left(  n\sum\limits_{x_{i}\in A_{n}}\!\!\mu
(S_{i})\right)  \leq\sum\limits_{n=1}^{\infty}\varepsilon\,2^{-n-2}
=\varepsilon/4,
\]
and the integral of $\left\Vert f\right\Vert $ over $\cup_{i}S_{i}$ is at most
$\varepsilon/4$. Set $B(0,-1)=B(0,0)=\varnothing$. If for $n\in\mathbb{N}$,
$x\in L\cap\left(  B(0,n)\diagdown B(0,n-1)\right)  $, then $B(x,1)\subseteq
E_{n}:=B\left(  0,n+1\right)  \diagdown B\left(  0,n-2\right)  $, and we
choose $\delta(x)$ with $0<\delta(x)\leq1$ so that every $\delta$-fine
$S\in\mathcal{S}$ with tag $x$ satisfies the inequality
\[
\int\nolimits_{S}\!\!\left\Vert f(y)-f(x)\right\Vert \,\mu(dy)<\frac
{\varepsilon\cdot2^{-n-2}}{\left[  1+\mu\left(  E_{n}\right)  \right]  }
\cdot\,\mu(S).
\]
With this choice of the gauge $\delta:X\rightarrow(0,1]$, we let $\left\langle
S_{i}\right\rangle $ be any finite or countably infinite sequence of $\delta
$-fine sets from $\mathcal{S}$ that is $\mu$-exhausting of $X$. Let $I_{L}$ be
the set of those indices $i$ for which $x_{i}\in L$, and let $I_{A}$ be the
set of those indices $i$ with $x_{i}\in A=X\setminus L$. For each
$n\in\mathbb{N}$, let $I_{L}^{n}$ be those indices in $I_{L}$ with $x_{i}\in
B(0,n)\backslash B(0,n-1)$. Set $S_{A}:=\cup_{i\in I_{A}}S_{i}(x_{i})$ and
$S_{L}:=\cup_{i\in I_{L}}S_{i}(x_{i})$. Now by the above calculation,
$\sum\limits_{i\in I_{A}}\left\Vert f(x_{i})\right\Vert \,\mu(S_{i}
)\leq\varepsilon/4$, and since $\mu\left(  X\backslash(S_{L}\cup
S_{A})\right)  =0$, we have
\[
\int_{X\backslash(S_{L}\cup S_{A})}\left\Vert f\right\Vert ~d\mu+\int_{S_{A}
}\left\Vert f\right\Vert ~d\mu+\sum\limits_{i\in I_{A}}\left\Vert
f(x_{i})\right\Vert \,\mu(S_{i})\leq\frac{\varepsilon}{2}.
\]
Moreover,
\begin{align*}
&  \int_{X}\left\Vert f(y)-\sum\nolimits_{i}\,f(x_{i})\chi_{S_{i}
}\,(y)\right\Vert ~\mu\left(  dy\right) \\
&  \leq\int_{S_{L}}\left\Vert f(y)-\sum\nolimits_{i\in I_{L}}\,f(x_{i}
)\chi_{S_{i}}\,(y)\right\Vert ~\mu(dy)+\frac{\varepsilon}{2}\\
&  \leq\sum_{n=1}^{\infty}\sum_{i\in I_{L}^{n}}\int_{S_{i}}\!\!\left\Vert
f(y)-f(x_{i})\right\Vert ~\mu(dy)+\frac{\varepsilon}{2}\\
&  <\sum_{n=1}^{\infty}\sum_{i\in I_{L}^{n}}\frac{\varepsilon\cdot2^{-n-2}
}{\left[  1+\mu\left(  E_{n}\right)  \right]  }\cdot\,\mu(S(x_{i}
))+\frac{\varepsilon}{2}\leq\varepsilon\cdot\sum_{n=1}^{\infty}2^{-n-2}
+\frac{\varepsilon}{2}<\varepsilon.
\end{align*}

To show that the sum of local errors is small, we note that for each $i$,
$\sum\nolimits_{j}\,f(x_{j})\chi_{S_{j}}(y)=f(x_{i})$ on $S_{i}$, whence
\begin{align*}
&  \sum\nolimits_{i}\left\Vert \int_{S_{i}}f(y)~\mu(dy)-f(x_{i})\mu
(S_{i})\right\Vert \leq\sum\nolimits_{i}\int_{S_{i}}\left\Vert f(y)-f(x_{i}
)\right\Vert ~\mu(dy)\\
&  =\sum\nolimits_{i}\int_{S_{i}}\left\Vert f(y)-\sum\nolimits_{j}
\,f(x_{j})\chi_{S_{j}}(y)\right\Vert ~\mu(dy)\\
&  =\int_{X}\left\Vert f(y)-\sum\nolimits_{j}\,f(x_{j})\chi_{S_{j}
}(y)\right\Vert ~\mu(dy)<\varepsilon.
\end{align*}

For the last inequality, choose $m\in\mathbb{N}$ so that
$\mu(\Omega \backslash\cup_{i\leq m}S_{i})<\gamma$,
and the proof follows essentially as
before. $\ \square$

\begin{remark}
\begin{rm}
Note that our simple functions are not formed using a partition of $X$, as is
usual for the Henstock--Kurzweil and McShane integrals; they are, however,
defined on all of $X$. Also note that partitions are allowed in Theorem
\ref{integration}. That is, suppose a Cartesian coordinate system has been
imposed on $X$ and $I\subset X$ is a bounded open interval. In McShane
integration over $I$, one takes a partition
$\left\langle S_{i}\right\rangle_{i=1}^{N}$ to be intervals in $I$ with
$\cup_{i=1}^{N}S_{i}=I$. If these
intervals are mutually disjoint, satisfy the $\lambda$-regularity condition,
and are $\delta$-fine, then they are examples of the covering sets in Theorem
\ref{integration}.
\end{rm}
\end{remark}

\begin{definition}
A function $G:\mathcal{M}\rightarrow Y$ is called \textbf{countably additive} if
$G(\cup_{i}E_{i})=\lim_{n\rightarrow\infty}\sum_{i}^{n}G(E_{i})$ in norm for
every ordering of any infinite, pairwise disjoint sequence
$\langle E_{i}\rangle$ from $\mathcal{M}$. It is also required that
$G(\varnothing)=0$, whence $G$ is finitely additive.
\end{definition}

\begin{corollary}
\label{sumimpliesintegrability}Let $\mu$ be a Radon measure on
$(X,\mathcal{M})$. Let $\mathcal{S}$ be a $\mu$-a.e.,
$\lambda$-Morse cover of $X$ such that
for each $S\in\mathcal{S}$, $\ \mu(S\diagdown S^{\circ})=0$, where $S^{\circ}$
is the interior of $S$. Suppose $G:\mathcal{M}\rightarrow Y$ is countably
additive and has the additional properties that $G(E)=0$ for each
$E\in\mathcal{M}$ with $\mu(E)=0$, and
\[
M:=\sup\left\{  \sum_{i}\left\Vert G(S_{i})\right\Vert :\left\langle
S_{i}\right\rangle \subset\mathcal{S}\text{,\ \ }\left\langle S_{i}
\right\rangle \text{\ }\mu\text{-exhausting of }X\right\}  <+\infty.
\]
Also suppose that there is a $\mu$-measurable $f:X\rightarrow Y$ such that for
any $\varepsilon>0$ there is a gauge $\delta:X\rightarrow(0,1]$ \ with the
property that for any $\delta$-fine sequence
$\left\langle S_{i}(x_{i})\right\rangle $ in $\mathcal{S}$
that is $\mu$-exhausting of $X$ we have
\begin{equation}
\sum\nolimits_{i}\left\Vert f(x_{i})\mu(S_{i})-G(S_{i})\right\Vert
<\varepsilon. \label{riemannsum}
\end{equation}
Then $f$ is $\mu$-integrable and $\int_{X}f\,d\mu=G(X)$.
\end{corollary}

\noindent \textbf{Proof:} Fix $\varepsilon>0$. Also, fix
$\langle T_{j}\rangle\subset\mathcal{S}$
so that $\langle T_{j}\rangle$ is a sequence that is $\mu$-exhausting of $X$
and $\sum_{j}\Vert G(T_{j})\Vert>M-\varepsilon/2$. Let
$\Omega=\cup_{j} T_{j}^{\circ}$. We choose a gauge $\delta:X\rightarrow(0,1]$
so that\ for any $\delta$-fine sequence $\left\langle S_{i}\right\rangle $ in
$\mathcal{S}$ that is $\mu$-exhausting of $X$, Equation \ref{riemannsum} holds with
$\varepsilon$ replaced by $\varepsilon/2$ \ and if $x\in T_{j}^{\circ}$ for
some $j$, then $B(x,\delta(x))\subseteq T_{j}^{\circ}$. The collection
$\mathcal{S}_{\Omega}$ of $\delta$-fine sets in $\mathcal{S}$ with tags in
$\Omega$ is a $\mu$-a.e., Morse cover of $\Omega$. Let
$\left\langle S_{i}\right\rangle $ be a $\delta$-fine sequence in $\mathcal{S}_{\Omega}$
such that $\left\langle S_{i}\right\rangle $ is $\mu$-exhausting of $\Omega$
and therefore of $X$. We have for any ordering,
\begin{align*}
\left\vert \sum_{i}\Vert f(x_{i})\Vert\mu(S_{i})-\sum_{i}\Vert G(S_{i}
)\Vert\right\vert  &  \leq\sum_{i}\left\vert \ \Vert f(x_{i})\mu(S_{i}
)\Vert-\Vert G(S_{i})\Vert\right\vert \\
&  \leq\sum_{i}\left\Vert f(x_{i})\mu(S_{i})-G(S_{i})\right\Vert
<\frac{\varepsilon}{2}.
\end{align*}
Moreover, for each $j$,
$\mu\left(  T_{j}\diagdown
{\textstyle\bigcup_{S_{i}\subseteq T_{j}}} S_{i}\right)  =0$, so
\begin{align*}
M  &  \geq\sum_{i}\Vert G(S_{i})\Vert=\sum_{j}\sum_{S_{i}\subseteq T_{j}}\Vert
G(S_{i})\Vert\\
&  \geq\sum_{j}\left\Vert G\left(
{\textstyle\bigcup_{S_{i}\subseteq T_{j}}}
S_{i}\right)  \right\Vert =\sum_{j}\left\Vert G\left(  T_{j}\right)
\right\Vert >M-\frac{\varepsilon}{2}.
\end{align*}
Therefore,
\[
\left\vert \sum_{i}\Vert f(x_{i})\Vert\mu(S_{i})-\ M\right\vert <\varepsilon.
\]
By Theorem~12 in \cite{loebtalvila}, $\Vert f\Vert$ is $\mu$-integrable on
$\Omega$, and therefore on $X$, whence $f$ is $\mu$-integrable on $X$.

To show that $\int_{X}f\,d\mu=G(X)$, we fix a gauge $\delta$ so that Equations
\ref{localerrorsum} and \ref{riemannsum} hold for any sequence
$\left\langle S_{i}(x_{i})\right\rangle $ of $\delta$-fine sets from
$\mathcal{S}$ that is
$\mu$-exhausting of $X$. Then given such a sequence, we have
\begin{align*}
&  \left\Vert G(X)-\int_{X}f\,d\mu\right\Vert \\
&  =\left\Vert \sum_{i}\left(  G(S_{i})-\int_{S_{i}}f\,d\mu\right)
\right\Vert \leq\sum_{i}\left\Vert G(S_{i})-\int_{S_{i}}f\,d\mu\right\Vert \\
&  \leq\sum_{i}\left\Vert \int_{S_{i}}f\,d\mu-f(x_{i})\mu(S_{i})\right\Vert
+\sum_{i}\left\Vert f(x_{i})\mu(S_{i})-G(S_{i})\right\Vert <2\varepsilon.
\end{align*}
Since $\varepsilon$ is arbitrary, $\int_{X}f\,d\mu=G(X)$. $\square$


TITLE: Continuity of a function $f : \mathbb R^n \to \mathbb R^k$
QUESTION [3 upvotes]: So preparing for the end-term of the semester, strolling trough exercises in our huge book (Th. Rassias Calculus 2) and in the internet, I found one more theoritical one, that as always bothers me (because it's theoritical !!).
Show that in the function $f : \mathbb R^n \to \mathbb R^k$ , $f=(f_1,\dots,f_k)$ is continuous at $\vec x_o$, if and only if the function $f_i$ is continuous at $\vec x_o$ for $i = [1,2,\dots,n]$.
Now, it's pretty obvious that since the "big" function $f$ is formed by the "smaller" $f_i$'s , all the sub-functions must be continuous, but that's not a mathematical proof of course. Also, I could say that for $f$ to be continuous, its limit must be defined, which means that $\lim_{x \to x_o}f = f_o$, which would mean that : $\lim_{x \to x_o}f = (\lim_{x \to x_o}f_1^o,\dots,\lim_{x \to x_o}f_k^o) = f_o = (f_1^o,\dots,f_k^o)$. Now, by definition that means that each of the sub-functions must be continuous, which means that they are continuous if and only if $\forall e>0$ $\exists d>0$ : $\forall x \in \mathbb R^n$ with distance $d(x,x_o)<d \Rightarrow d(f(x),f(x_{io})<d$. Is that a kind of correct claim ? I am pretty sure it's not as mathematical it should be. Any tips-hints-corrections or help over this will be really appreciated !

REPLY [0 votes]: $(\Leftarrow)$ Suppose $f$ is continuous, you could use the fact that composition of continuous functions is continuous to prove that all of the $f_i\text{'s}$ are continuous as well..
Define the $i\text{-th}$ coordinate projection function $$\pi_i:\mathbb{R}^k \rightarrow \mathbb{R}\\(x_1,\ldots,x_k)\mapsto x_i,\quad i\in\{1,\ldots,k\}$$
which is linear and hence continuous. Then,
$$f_i=\pi_i \circ f,\quad i\in\{1,\ldots,k\} $$
are continuous as compositions of continuous functions.
$(\Rightarrow)$ For the other direction, continuity of the $f_i\text{'s}$ implies,
$$\forall \epsilon>0. \exists \delta>0:(\Vert x-x_0\rVert<\delta)\implies\lVert(f_i(x)-f_i(x_0)\rVert<\epsilon$$
Taking the supremum norm in $\mathbb{R}^k$, defined as $\lVert y\rVert_{\infty}=\sup_k\{\lvert y_k\rvert\}, y\in \mathbb{R}^k$, we see that point-wise convergence of the $f_i\text{'s}$ suggests convergence in $\mathbb{R}^k$.


TITLE: Remainder when $N$ is divided by '$9$'
QUESTION [1 upvotes]: $N$ is a natural number formed by writing, in the ascending order, the first $1002$ whole numbers one after the another. Find the remainder when $N$ is divided by $9$
It is known that if the summation of digits of $N$ is divisible by $9$, then $N$ is. Otherwise, summation when done recursively until it can be divided by $9$ easily, remainder then obtained will be the remainder when $N$ is divided by $9$. I can't figure out how to find summation of the digits of the number $N$.

REPLY [0 votes]: As far as 9 is concerned, there’s no difference between adding numbers and adding up all their digits. For example, $24+39=63$ and $2+4+3+9=18$; either way, it’s a multiple of $9$.
Now the sum $1+\cdots +9$ is a multiple of $9$. So is $10+\cdots +18$. Indeed, for any $k$, we have that $(9k+1)+\cdots +(9k+9)$ is a multiple of $9$, and therefore, so is its digit sum. This applies to every run of $9$ in your total. The last multiple of $9$ under $1002$ is $999$, so you really only need to check the numbers past that point:
$$1+0+0+0+1+0+0+1+1+0+0+2=6$$