text:
\begin{document}
\title[Measures and Processes on Semilattices]{Measures and Processes on Semilattices}
\myself \date \today \subjclass[2000]{Primary: 28A05, 60G05.}
\keywords{Lattice, Modular set functions, Extension of measures, Stochastic Integration, Stochastic
Processes, Doob-Meyer Decomposition.}

\begin{abstract}
Every vector-valued, semi-modular set function on a semi-lattice of sets extends uniquely to an
additive set function on the generated ring. Based on this theorem we outline a theory of
stochastic processes when the underlying index set is a semi-lattice and obtain a characterization
of processes as vector measures and a form of the Doob-Meyer decomposition.
\end{abstract}

\maketitle

\section{Introduction and Notation}

The need that often arises in stochastic analysis to examine the properties of processes
along the hitting times of some family of subsets of $\R$, spurs interest for a theory of
stochastic processes with an index set more general than the time domain. The history of
such a theory is, of course, a long one, but most authors incline to credit Cairoli and Walsh
\cite{cairoli walsh} for having given start to it. A rather complete list of contributions, which
is beyond scope here, may be found in the bibliographic references of the book of Ivanoff
and Merzbach \cite{ivanoff merzbach}.

In the present paper we work with an index set $T$ which has the only properties of being
partially ordered and closed with respect to one lattice operation, either $\vee$ or $\wedge$:
a semi-lattice. The explicit aim is proving some of the most useful results of the classical theory
of processes, such as the decomposition of Doob and Meyer, and see how much of that theory
carries over to the more general setting adopted here. This attempt, that one should confront
with a preceding paper \cite{SPTA}, is made while trying to resist the temptation of restoring
classical properties via additional \textit{ad hoc} assumptions. Thus, e.g., $T$ will have no
topological structure at all.

The problem considered here leads quite naturally to look for a convenient extension of the
given index set. Proving the existence of a desirable extension has a direct translation into
the somehow more general problem of extending set functions which are originally defined
on a class of sets that is closed under only one set operation, either $\cap$ or $\cup$, a semi-
lattice of sets. The existence of a measure theoretic counterpart to whatever a theory of
stochastic processes comes, of course, at no surprise given the many crucial papers focusing
exactly on this link, among which the important one of Dol\'{e}ans-Dade \cite{doleans-dade}
and even the original contribution \cite{ito} of \^{I}to. A main source of inspiration for this
paper has however been the unpublished work of Norberg \cite{norberg}.

The present paper deals in section 2 with the measure theoretic problem described above. We
provide an exact characterization which is then easily extended to product spaces. This last result
translates immediately, in section 5, into the statement that stochastic processes indexed by
semilattices may be regarded as vector-valued, additive set functions. Eventually, in sections
5.1 and 5.2 we prove some classical results such as the Riesz decomposition for quasi
martingales and a version of the decomposition of Doob and Meyer. To obtain the latter
decomposition, we exploit a fairly simple, weak compactness condition, the class $DM$
property, which is shown to be nothing but an extension of the more classical class $D$ property
originally introduced by Meyer \cite{meyer}. It is shown, however, that the predictability property
of the intervening compensator is in the present context more troublesome from the point of view
of interpretation although quite easy mathematically. In particular, there is a multiplicity of possible
definitions of predictability for the compensator, although, given an arbitrary choice among these,
the decomposition is unique.

As in \cite{SPTA} we find convenient the following notation. When $S$ is a set, $2^S$ and
$\set{S}$ indicate the collection of all of its subsets and its indicator function. If
$\Sigma\subset2^S$, then $\Sim(\Sigma)$ designates the collection of $\Sigma$ simple
functions on $S$, $\B(\Sigma)$ its closure with respect to the topology of uniform convergence
and $ba(\Sigma)$ (resp. $ca(\Sigma)$) the family of bounded, finitely (resp. countably) additive
set functions on $\Sigma$. In general we omit to refer to $\Sigma$ whenever $\Sigma=2^S$. If
$\lambda$ is a set function on $\Sigma$, the $\lambda$ integral of $f$, if well defined, is denoted
by $\lambda(f)$ or $\int fd\lambda$, according to notational convenience.

If $N\in\N$ and $a\subset\N$ we shall write $[N]=\{1,\ldots,N\}$ and denote by $\abs a$ the
cardinality of $a$; moreover, if $b\subset\N$ the lattice notation $a\le b$ is preferred to
$a\subset b$ and $a<b$ signifies that $a$ is a strict subset of $b$. Write
$\mathcal N=\{a\subset\N:\abs a<\infty\}$ and,
\begin{equation}
\label{nu}
\nu(b)=(-1)^{1+\abs b}\qquad b\in\mathcal N
\end{equation}
If $a\le b$, one easily concludes that $\nu(b)=\mu(a,b)\nu(a)$ where $\mu$ is the M\"{o}bius
function on $\mathcal N$, see \cite[Corollary, p. 345]{rota}. We thus deduce from
\cite[Propositions 1 and 2, p. 344]{rota} that $\nu$ is the unique set function on $\mathcal
N$ satisfying satisfying: (\textit{i})
\begin{equation}
\label{Mobius function}
\nu(\emp)=-1\quad\text{and}\quad\sum_{a\le x\le b}\nu(x)=0
\qquad a,b\in\mathcal N,\ a<b
\end{equation}
and (\textit{ii}) if $f:\mathcal N\to\R$ and $F(a,b)=\sum_{\{y:a\le y\le b\}}f(y)$
then the following inversion formulas hold
\begin{equation}
\label{Mobius inversion}
\nu(b)f(b)=\sum_{\{y:a\le y\le b\}}F(a,y)\nu(y)\quad \text{and}\quad
\nu(a)f(a)=\sum_{\{y:a\le y\le b\}}F(y,b)\nu(y)
\end{equation}
As an illustration take $\{A_1,\ldots,A_N\}\subset X$, $b\le[N]$ and
$A_b=(\bigcap_{n\in b}A_n^c)\cap(\bigcap_{k\in b^c\cap[N]}A_k)$ (with
$\bigcap_{n\in\emp}A_n=X$).
The equality $\set{\bigcap_{n\in a}A_n^c}=\sum_{b\ge a}\set{A_b}$, \eqref{Mobius function}
and \eqref{Mobius inversion} imply
\begin{equation}
\label{lattice representation}
\set{\bigcap_{n\in [N]}A_n}=-\nu(\emp)\set{A_\emp}
=-\sum_{\emp\le b\le [N]}\nu(b)\set{\bigcap_{n\in b}A_n^c}
=\sum_{\emp\le b\le [N]}\nu(b)\set{\bigcup_{n\in b}A_n}
=\sum_{\emp<b\le [N]}\nu(b)\set{\bigcup_{n\in b}A_n}
\end{equation}
and likewise
$\set{\bigcup_{n\in [N]}A_n}=\sum_{\emp<b\le [N]}\nu(b)\set{\bigcap_{n\in b}A_n}$.

\section{semi-modular Set Functions}

In this section $X$ will be an arbitrary set and $Y$ a real vector space with neutral element
denoted by $\0$. $\A\subset2^X$ is said to be a lattice of sets if it is non empty and closed
with respect to $\cap$ and $\cup$\footnote{Some authors, e.g. Halmos \cite[p. 25]{halmos}
and Bogachev \cite[p. 75]{bogachev}, include the property $\0\in\A$ in the definition of a
lattice.}. Several authors, including Bachman and Sultan \cite{bachman sultan} and Bhaskara
Rao and Bhaskara Rao \cite{rao}, have studied set functions on lattices of sets and the possibility
of extending them to a larger class, typically the generated algebra or ring. This possibility hinges
on the two properties introduced next.

We speak of $\Phi:\A\to Y$ as a modular set function whenever $\A$ is a lattice of sets
and\footnote{Pettis \cite{pettis} calls this property $2$-additivity.}
\begin{equation}
\label{modular}
 \Phi(A\cup B)+\Phi(A\cap B)=\Phi(A)+\Phi(B)\qquad A,B\in\A
\end{equation}
Simple induction shows that \eqref{modular} is equivalent to either one of the following
two properties\footnote{Chung \cite[p. 329]{chung} credits Poincar\'{e} for being the first to obtain
\eqref{modular cap}.}:
\begin{subequations}
\label{semi-modular}
\begin{equation}
\label{modular cap}
\Phi\left(\bigcup_{n\in[N]}A_n\right)
=\sum_{\emp<b\le[N]}\nu(b) \Phi\left(\bigcap_{n\in b}A_n\right)
\qquad A_1,\ldots,A_N\in\A
\end{equation}
\begin{equation}
\label{modular cup}
\Phi\left(\bigcap_{n\in[N]}A_n\right)
=\sum_{\emp<b\le[N]}\nu(b)\Phi\left(\bigcup_{n\in b}A_n\right)
\qquad A_1,\ldots,A_N\in\A
\end{equation}
\end{subequations}

The property  $\Phi(\emp)=\0$ whenever $\emp\in\A$ does not follow from \eqref{modular}
and we then say that $\Phi:\A\to Y$ is strongly additive, in symbols $\Phi\in sa(\A,Y)$, if it
is modular and satisfies
\begin{equation}
\label{sa}
\text{either}\qquad\emp\notin\A\quad\text{or}\qquad\Phi(\emp)=\0
\end{equation}
In other words, $\Phi\in sa(\A,Y)$ if and only if it admits an extension
$\bar\Phi\in sa(\A\cup\{\emp\},Y)$. It is also easy to see that each $\Phi\in sa(\A,Y)$ is additive
and that, if $\A$ is a ring, then the converse is also true.

The case of modular or strongly additive set functions on lattices of sets was studied in depth by
Pettis \cite{pettis}. We are interested in defining similar properties when the underlying family
$\A_0\subset2^X$ is just a semi-lattice, i.e. a collection of sets closed with respect to just one set
operation, either $\cap$ or $\cup$. Important examples of semilattices are, of course, ideals
and filters; a $\cap$-lattice is also known in probability as a $\pi$ system\footnote{A $\cap$
-lattice containing the empty set was called \textit{intersectional class} by Groemer
\cite[p. 397]{groemer}}. $\Phi:\A_0\to Y$ is then said to be semi modular if either (\textit{i})
$\A_0$ is a $\cap$-lattice and \eqref{modular cap} holds for every finite collection
$A_1,\ldots,A_N\in\A_0$ such that $\bigcup_{n=1}^NA_n\in\A_0$ or (\textit{ii}) $\A_0$ is
a $\cup$-lattice and \eqref{modular cup} holds for every finite collection $A_1,\ldots,A_N\in\A_0$
such that $\bigcap_{n=1}^NA_n\in\A_0$.

There exists of course a perfect symmetry between these two properties.

\begin{lemma}
\label{lemma conjugate}
Let $\A_0\subset 2^X$ and $y\in Y$. Then, (i) $\A_0$ is a semi-lattice if and only if so is
$\A_0^c=\{A^c:A\in\A_0\}$; (ii) $\Phi:\A_0\to Y$ is semi modular if and only if
so is its conjugate and its $y$ translation, i.e. the set functions $\Phi^c:\A^c\to Y$ and
$\Phi_y:\A\to Y$ defined implicitly via
\begin{align}
\label{conjugate}
\Phi^c(A^c)&=-\Phi(A)&\Phi_y(A)&=\Phi(A)+y& A\in\A_0
\end{align}

\end{lemma}
\begin{proof}
The first claim is obvious. Assume that $\A_0$ is a $\cap$-lattice and $\Phi:\A_0\to Y$ is
semi-modular. If $A_1\ldots,A_N,\bigcup_{n=1}^NA_n\in\A_0$, then,
we conclude
\begin{align*}
\Phi^c\left(\bigcap_{n=1}^NA^c_n\right)&=\Phi\left(\bigcup_{n=1}^NA_n\right)
=\sum_{\emp<b\le[N]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)
=\sum_{\emp<b\le[N]}\nu(b)\Phi^c_y\left(\bigcup_{n\in b}A^c_n\right)
\end{align*}
The fact that $\Phi_y$ is semi-modular follows from \eqref{Mobius function}.
The poof for the case in which $\A_0$ is a $\cup$-lattice is identical.
\end{proof}
Lemma \ref{lemma conjugate} allows to restrict our proofs to the case of $\cap$-lattices.

Set functions on semilattices, even if not semi-modular, satisfy some minimal properties.
\begin{lemma}
\label{lemma pre norberg}
Let $\A_0$ be a $\cap$-lattice, $A_1,\ldots,A_N\in\A_0$ and $\Phi:\A_0\to Y$. The following is
true:
\begin{equation}
\label{decomposition}
\sum_{\0<b\le[N]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)
=\Phi(A_N)+\sum_{\0<b\le[N-1]}\nu(b)
\left[\Phi\left(\bigcap_{n\in b}A_n\right)-\Phi\left(A_N\cap\bigcap_{n\in b}A_n\right)\right],
\end{equation}
\begin{equation}
\label{parsimony}
A_N\subset A_{N-1}\quad\text{implies}\quad
\sum_{\0<b\le[N]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)=
\sum_{\0<b\le[N-1]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)
\end{equation}
\eqref{decomposition} and \eqref{parsimony} remain true after exchanging $\cap$ with $\cup$ and
reversing inclusions.
\end{lemma}

\begin{proof}
\eqref{decomposition} is a consequence of
\begin{align*}
\sum_{\0<b\le[N]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)&=
\sum_{\0<b\le[N-1]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)
+\sum_{\{N\}\le b\le[N]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)\\
&=\sum_{\0<b\le[N-1]}\nu(b)\Phi\left(\bigcap_{n\in b}A_n\right)
-\sum_{\0\le b\le[N-1]}\nu(b)\Phi\left(A_N\cap\bigcap_{n\in b}A_n\right)\\
\end{align*}
and implies that $\sum_{\0<b\le[N-1]}\nu(b)\Phi\left(A_N\cap\bigcap_{n\in b}A_n\right)$
may equivalently be written as
\begin{align*}
\Phi(A_N\cap A_{N-1})+\sum_{\0<b\le[N-2]}\nu(b)\left[\Phi\left(A_N\cap\bigcap_{n\in b}A_n\right)
-\Phi\left(A_N\cap A_{N-1}\cap\bigcap_{n\in b}A_n\right)\right]
\end{align*}
or, if $A_N\subset A_{N-1}$, as $\Phi(A_N)$. \eqref{parsimony} is then immediate. The last claim
is clear.
\end{proof}

For some semilattices all set functions $\Phi:\A_0\to Y$ are semi-modular.
\begin{corollary}
\label{cor pre norberg}
Assume that $\A_0$ is a $\cap$-lattice with the following property:
\begin{equation}
\label{property}
A_1,\ldots,A_N,\bigcup_{n=1}^NA_n\in\A_0\text{  implies  }\bigcup_{n=1}^NA_n=A_{n_0}
\text{ for some }1\leq n_0\leq N
\end{equation}
Then, every $\Phi:\A_0\to Y$ is semi-modular.
\end{corollary}
\begin{proof}
This is a straightforward consequence of \eqref{parsimony}.
\end{proof}

{Property \eqref{property} appears as an assumption in \cite[A1, p. 502]{dozzi} and
\cite[Assumption 1.1.5, p. 13]{ivanoff merzbach} but it is in fact a natural condition in
order lattices, see section 3.

We generalize the notion of a strongly additive set function given above to that of a semi-additive
set function by saying that $\Phi:\A_0\to Y$ is semi-additive, in symbols $\Phi\in sa_0(\A_0,Y)$,
if it may be extended as a semi-modular set function to the semi-lattice $\A_0\cup\{\emp\}$ with
$\Phi(\emp)=\0$. Necessary and sufficient conditions for the existence of such an extension
depend on the structure of $\A_0$. If $\A_0$ is $\cap$ closed $\Phi$ is semi-additive if and only
if it satisfies \eqref{sa}; if $\A_0$ is $\cup$ closed, semi-additivity is equivalent to
\begin{equation}
\label{sa cup}
\sum_{\emp< b\le[N]}\nu(b)\Phi\left(\bigcup_{n\in b}A_n\right)=\0
\qquad A_1,\ldots,A_N\in\A_0,\ \bigcap_{n=1}^NA_n=\emp
\end{equation}
In either case a semi-modular set function on a semi-lattice is semi-additive whenever
$\bigcap_{n=1}^NA_n\neq\emp$ for all $A_1,\ldots,A_N\in\A_0$, again a property
assumed in \cite[Definition 1.1.1, p. 10]{ivanoff merzbach} or \cite[A3, .p. 63]{saada slonowsky}.

We end this section with some examples of semi-modular set functions.
\begin{example}
Let $\A_0$ the collection of all the finite dimensional subspaces of a vector space $X$ and define
$\Phi:\A_0\to\R$ by letting $\Phi(A)=\dim(A)$. Then, $\A_0$ is a $\cap$-lattice and,
by Grasmann formula, $\Phi$ is $\cap$-modular.
\end{example}
\begin{example}
\label{ex dirac}
Let $\A_0$ be a $\cup$-lattice of subsets of some set $X$, $Y$ the collection of real-valued
functions on $X$ and define $\Phi:\A_0\cup\{\emp\}\to Y$ by letting $\Phi(A)=\set{A}$.
Then, \eqref{lattice representation} implies that $\Phi$ is semi-modular. The same is true of
$\delta_x$, the Dirac measure sitting at $x$, for all $x\in X$.
\end{example}

\begin{example}
Let $\A_0$ be the collection of all compact, finite-dimensional polytopes of some vector
space $X$, a $\cap$-lattice. More explicitly, each $A\in\A_0$ may be written in the
form $A=\co\{x_1,\ldots,x_k\}$ for a minimal (and therefore unique) set $\{x_1,\ldots,x_k\}$
of its vertices so that $\dim(A)=k$. Equivalently, we have the hyperplane representation
$A=\{x\in X:Tx\ge b\}$ where $T$ is a linear map from $X$ to $\R^d$ and $b\in\R^d$
for some $d>0$. Let $\Phi:\A_0\to Y$ be given. If $A_1,\ldots,A_N\in\A_0$ and
$A=\bigcup_{n=1}^NA_n\in\A_0$ then $\dim(A_N)<\dim(A)$  implies
$A_N\subset\bigcup_{n=1}^{N-1}A_n$. By compactness there would otherwise exist an open
set $O\subset X$ such that $A_N\cap O\neq\emp=O\cap\bigcup_{n=1}^{N-1}A_n$.
But then $A_N\cap O$ and $A\cap O$ coincide despite having different dimension. Choose
$d$ to be the minimal linear dimension for which there is a collection $\{A_1,\ldots,A_N\}$
in $\A_0$ with $A=\bigcup_{n=1}^NA_n\in\A_0$, $\dim(A)=d$ but for which
\eqref{modular cap} fails; choose $N$ to be the minimal number of sets for which
this may occur. Of course $d>0$ and $N>1$. Since $N$ is minimal, Lemma
\ref{lemma pre norberg} implies that all sets $A_n$ must have the same dimension, $d$.
Let $\mathscr H$ be the collection of all hyperplanes in $X$ and, for each $H\in\mathscr H$,
let $H^+,H^-$ be the corresponding half-spaces. Groemer \cite{groemer} considered the
following property
\begin{equation}
\label{groemer}
\Phi(A)=\Phi(A\cap H^+)+\Phi(A\cap H^-)-\Phi(A\cap H)\qquad A\in\A_0,\ H\in\mathscr H
\end{equation}
Assume \eqref{groemer}, choose $H\in\mathscr H$ so that $H^+$ contains $A_1$ but not $A$
and let $A_n^+=A_n\cap H^+$, $A_n^-=A_n\cap H^-$ and $A_n^H=A_n\cap H$. By
assumption, $\dim(A^H)<d$ and $\dim(A_1^-)=\dim(A_1^H)<\dim(A^-)$ so that
$A^-=\bigcup_{n=2}^NA_n^-$. Given that $d$ and $N$ were chosen to be minimal,
$\{A_1,\ldots,A_N\}$ fails to satisfy \eqref{modular cap} if and only if so does
$\{A_1^+,\ldots,A_N^+\}$. Repeating this reasoning for all $H\in\mathscr H$ with
$A_1\subset H^+$ leads to the conclusion that the collection
$\{A_1,A_2\cap A_1,\ldots,A_1\cap A_N\}$ does not meet \eqref{modular cap}, a
contradiction of Lemma \ref{lemma pre norberg}. Thus,
\begin{lemma}[Groemer]
\label{lemma groeber}
If $X$, $\A_0$ and $\Phi$ are as above and $\Phi$ satisfies \eqref{groemer},
then, $\Phi$ is semi-modular.
\end{lemma}
\noindent Klain and Rota \cite{klain rota} show that all set functions assigning to a compact
polytope some polynomial of the length of its faces actually satisfy \eqref{groemer}.
\end{example}

\section{Additive Extensions}
The main result of this section is a necessary and sufficient condition for extending semi-modular,
vector-valued set functions from a semi-lattice of sets to the generated ring. The measure
extension problem is, of course, a time honored one and, among the many contributions, a
major one has been that of Horn and Tarski \cite[theorem 1.21, p. 477]{horn tarski} based on
the notion of partial measure on which we shall briefly comment later on. A general treatment
of this topic following these same guidelines is contained in \cite[chapter 3]{rao}; see also
\cite{konig}.

Given their repeated use, we state hereafter some well known facts.

\begin{lemma}
\label{lemma semi-lattice}
Let $\A_0\subset2^X$ be a $\cup$- (resp. $\cap$-) lattice and $\A_1$ and $\A$ the lattice and
the ring generated by $\A_0\cup\{\emp\}$, respectively. $\A_1$ consists of finite (possibly
empty) intersections (resp. unions) of sets from $\A_0$ while each $A\in\A$ is the disjoint
union $\bigcup_{n=1}^NB_n\backslash C_n$ where $B_n,C_n\in\A_1$ and $C_n\subset B_n$
for $n=1,\ldots,N$. Moreover, $\Sim(\A_0)=\Sim(\A)$.
\end{lemma}

\begin{proof}
The representation of the members of $\A_1$ is an easy fact; for $\A$ see \cite[p. 26]{halmos}
or \cite[theorem 1.1.9, p. 4]{rao}. From \eqref{lattice representation} one easily establishes
$\Sim(\A_1)=\Sim(\A_0)$. Given the representation $\bigcup_{n=1}^NB_n\cap C_n^c$ of
each $A\in\A$ as in the claim, we obtain $\set{A}=\sum_{n=1}^N(\set{B_n}-\set{C_n})$ so
that $\Sim(\A)=\Sim(\A_1)$.
\end{proof}

The following is the main result of this section.

\begin{theorem}
\label{th extension}
Let $\A_0\subset2^X$ be a semi-lattice and $\A_1$ the generated lattice.
$\Phi_0:\A_0\to Y$ is semi-modular if and only if there exists a necessarily unique,
modular set function $\Phi_1:\A_1\to Y$ such that $\lrestr{\Phi_1}{\A_0}=\Phi_0$.
\end{theorem}

\begin{proof}
By Lemma \ref{lemma conjugate} we just consider the case of a $\cap$-lattice.
If an extension such as $\Phi_1$ in the statement exists it must take the form
\begin{equation}
\label{k}
\Phi_1\left(\bigcup_{n=1}^NA_n\right)=
\sum_{\emp<b\le[N]}\nu(b)\Phi_0\left(\bigcap_{n\in b}A_n\right)
\qquad A_1,\ldots,A_N\in\A_0
\end{equation}
and is thus necessarily unique. Denote by $\lambda(\{A_1,\ldots,A_N\})$ the right hand side of
\eqref{k}. It is clear that $\lambda$ is invariant with respect to a permutation of the indexes.
Moreover, if $A_n,B_k\in\A_0$ for $n=1\ldots,N$ $k=1\ldots,K$ and
$\bigcup_{n=1}^NA_n=\bigcup_{k=1}^KB_k$  we know from \eqref{parsimony} that
$\lambda(\{A_1,\ldots,A_N\})=\lambda(\{A_1,\ldots,A_N,B_1,\ldots,B_K\})
=\lambda(\{B_1,\ldots,B_K\})$. Thus the right hand side of \eqref{k} indeed defines an extension
$\Phi_1:\A_1\to Y$ of $\Phi_0$. To prove that this is modular let $A=\bigcup_{n=1}^NA_n$ and
$B=\bigcup_{k=1}^KB_k$ be elements of $\A_1$, with $A_n,B_k\in\A_0$ for $n=1,\ldots,N$ and
$k=1,\ldots,K$. Define also the collection $\{C_m:m=1,\ldots,N+K\}$ with $C_m=A_m$ for
$m\leq N$ or $C_m=B_{m-N}$ when $N<m\leq N+K$. Then, since $\nu(a\cup b)=
\nu(a)\nu(b)$ when $a\cap b=\emp$,
\begin{align*}
\Phi_1(A\cup B)&=\Phi_1\left(\bigcup_{m=1}^{N+K}C_m\right)\\
&=\sum_{\emp<b\le[N+K]}\nu(b)\Phi_1\left(\bigcap_{n\in b}C_n\right)\\
&=\sum_{\emp<b\le[N]}\nu(b)\Phi_1\left(\bigcap_{n\in b}A_n\right)
+\sum_{\emp<a\le[K]}\nu(a)\Phi_1\left(\bigcap_{k\in a}B_{k}\right)\\
&\quad +\sum_{\emp<b\le[N],\emp<a\le[K]}\nu(a)\nu(b)
\Phi_1\left(\bigcap_{k\in a}\left(B_k\cap\bigcap_{n\in b}A_n\right)\right)\\
&=\Phi_1(A)+\Phi_1(B)-\sum_{\emp<b\le[N]}\nu(b)\Phi_1\left(\bigcap_{n\in b}(B\cap A_n)\right)\\
&=\Phi_1(A)+\Phi_1(B)-\Phi_1(A\cap B)
\end{align*}
Given that $\A_1$ is a lattice, this is equivalent to the statement that $\Phi_1$ is semi-modular.
\end{proof}

The following is a minor generalization of a classical result of Pettis
\cite[theorem 1.2, p. 188]{pettis}.

\begin{theorem}[Pettis]
\label{th pettis}
Let $\A_0$ be a semi-lattice of subsets of $X$, $\A$ the generated ring and let $\Phi_0:\A_0\to Y$.
$\Phi_0$ is semi-modular if and only if there exists a necessarily unique $\Phi\in sa(\A,Y)$ satisfying
\begin{equation}
\label{pettis}
\Phi(A\backslash B)=\Phi_0(A)-\Phi_0(B)\qquad A,B\in\A_0,\ B\subset A
\end{equation}
Moreover, (i) $\Phi$ extends $\Phi_0$ if and only if $\Phi_0\in sa_0(\A_0,Y)$ and (ii)
if $\bar\A$ is any algebra containing $\A_0$ then $\Phi_0\in sa(\A_0,Y)$ has an additive extension
to $\bar\A$ and this will be unique up to the choice of $\bar\Phi(X)$ if $\bar\A$ is the algebra
generated by $\A_0$.
\end{theorem}
\begin{proof}
If $\Phi_0$ is semi-modular let $\Phi_1$ be its modular extension to
the generated lattice $\A_1$. Let $\A_2=\{A\backslash B:A,B\in\A_1,\ B\subset A\}$,
a $\cap$-lattice. If $A\backslash B,\ C\backslash D\in\A_2$ and $A\backslash B=C\backslash D$
then, since $\Phi_1$ is modular, $\Phi_1(A\cup D)=\Phi_1(A)+\Phi_1(D)-\Phi_1(A\cap D)$
so that the equalities $A\cup D=C\cup B$ and $A\cap D=C\cap B$ imply
$\Phi_1(A)-\Phi_1(B)=\Phi_1(C)-\Phi_1(D)$. We are then free to define $\Phi_2:\A_2\to Y$ by
letting
\begin{equation}
\Phi_2(A\backslash B)=\Phi_1(A)-\Phi_1(B)\qquad A\backslash B\in\A_1
\end{equation}
Clearly $\emp\in\A_2$ and $\Phi_2(\emp)=\0$. By Lemma \ref{lemma semi-lattice}, $\A$ is
the lattice generated by $\A_2$. Suppose $A_n\backslash B_n,A\backslash B\in\A_2$ for
$n=1,\ldots,N$ and $\bigcup_{n=1}^NA_n\backslash B_n=A\backslash B$. Then,
$$A\backslash B=\bigcup_{n=1}^N\bar A_n\left\backslash\bigcap_{n=1}^N\bar B_n\right.
\quad\text{where}\quad
\bar A_n=B\cup(A\cap A_n),\ \bar B_n=B\cup(A\cap B_n)$$
Thus, $\bar A_n\backslash\bar B_n=A_n\backslash B_n$ and therefore
\begin{align*}
\Phi_2(A\backslash B)
&=\Phi_1\left(\bigcup_{n=1}^N\bar A_n\right)-\Phi_1\left(\bigcup_{n=1}^N\bar B_n\right)\\
&=\sum_{\emp<b\le[N]}\nu(b)\left[\Phi_1\left(\bigcap_{n\in b}\bar A_n\right)
-\Phi_1\left(\bigcup_{n\in b}\bar B_n\right)\right]\\
&=\sum_{\emp<b\le[N]}\nu(b)\Phi_1\left(\bigcap_{n\in b}(\bar A_n\backslash\bar B_n)\right)\\
&=\sum_{\emp<b\le[N]}\nu(b)\Phi_1\left(\bigcap_{n\in b}(A_n\backslash B_n)\right)
\end{align*}
Thus, $\Phi_2\in sa_0(\A_2,Y)$ and admits, by Theorem \ref{th extension}, an extension
$\Phi\in sa(\A,Y)$  satisfying \eqref{pettis}. Suppose that the same is true of $\Psi$ and fix
$A,B\in\A_0$. If $\A_0$ is a $\cap$-lattice, write
$
\Psi(A)-\Psi(A\cap B)=\Phi(A)-\Phi(A\cap B)
$
(otherwise $\Psi(A\cup B)-\Psi(B)=\Phi(A\cup B)-\Phi(B)$). Given that both $\Psi$ and $\Phi$
are modular on $\A_1$, this equality is easily extended first to all $B\in\A_1$ for fixed $A\in\A_0$
and then to all $A,B\in\A_1$. But then $\restr{\Psi}{\A_2}=\restr{\Phi}{\A_2}$. Uniqueness then
follows from Theorem \ref{th extension}. Claims (\textit{i}) and (\textit{ii}) are obvious.
\end{proof}

A first implication of Theorems \ref{th extension} and \ref{th pettis} is that if $\Phi_0:\A_0\to Y$
is a semi-modular set function then there always exist a semi-additive set function obtained
from it by translation (see \eqref{conjugate}).

\begin{corollary}
\label{corollary semiadditive}
Every semi-modular $\Phi_0:\A\to Y$ admits a unique semi-additive translation.
\end{corollary}

\begin{proof}
Let $\Phi_1$ be the modular extension of $\Phi_0$ to the generated lattice and write
$y=-\Phi_1(\emp)$, if $\emp\in\A_1$, or $y=\0$ otherwise and denote, as in \eqref{conjugate},
translations by subscripts. Then $\Phi_ {1,y}\in sa(\A_1,Y)$ by construction while
$\Phi_{0,y}\in sa_0(\A_0,Y)$ because of $\Phi_{0,y}=\restr{\Phi_{1,y}}{\A_0}$. If $\Phi_{0,z}$
were another semiadditive translation of $\Phi_0$ then, by Theorem \ref{th pettis}, $\Phi_{0,y}$
and $\Phi_{0,z}$ would have the same additive extension to $\A$. Thus, if $A\in\A_0$ one has
$0=\Phi_{0,y}(A)-\Phi_{0,z}(A)=y-z$.
\end{proof}

Theorem \ref{th pettis} also implies a finitely additive version of Dynkin's lemma:

\begin{corollary}
\label{corollary Dynkin}
Two finitely additive probabilities agree on a semi-lattice if and only if they agree on the generated
algebra.
\end{corollary}

The existence of an additive extension to the generated ring, established in Theorem \ref{th pettis}
for semi-additive set functions on a semi-lattice, allows to associate this class of set functions with
the family of linear functionals on the vector space of simple functions. It is easy to see that this
relationship is isomorphic and in principle may be exploited to obtain even more general measure
extensions. This approach was inaugurated by Horn and Tarski \cite[Definition 1.6, p. 471]{horn
tarski} who first introduced  the notion of \textit{partial measure}, later generalized to that of
\textit{real partial charge} in \cite[Definition 3.2.1, p. 64]{rao}. A real-valued set function
$\lambda$ on some, arbitrary collection $\mathscr B\subset 2^X$ is a real partial charge if
$\sum_{n=1}^N\set{C_n}=\sum_{k=1}^K\set{D_k}$ with $C_n,D_k\in\mathscr B$ implies
$\sum_{n=1}^N\lambda(C_n)=\sum_{k=1}^K\lambda(D_k)$ and, if so, it may be extended to
a real-valued, additive set function on any algebra containing $\mathscr B$,
\cite[Theorem 3.2.5, p. 65]{rao}. In other words real partial charges may be extended with no
restriction on the initial collection of sets, while we had to assume that $\A_0$ is a semi-lattice.
However, this condition belongs more to analysis than to measure theory and in several cases it
is rather difficult to apply. It turns however very useful to establish further properties such as
positivity or boundedness at least when $Y=\R$.

\begin{lemma}
\label{lemma positive}
Let $\A_0$ be a semi-lattice, $\Phi_0\in sa_0(\A_0)$ and $\Phi\in sa(\A)$ the unique extension
of $\Phi_0$ to the ring $\A$generated by $\A_0$. (\textit{i}) $\Phi\geq0$ if and only if
$\sup\{\Phi_0(b):0\geq b\in\Sim(\A_0)\}\leq0$, (\textit{ii}) $\Phi\in ba(\A)$ if and only if
$\sup\{\dabs{\Phi_0(b)}:1\geq\abs{b}\in\Sim(\A_0)\}<\infty$,
(\textit{iii}) $\Phi\in ba(\A)_+$ if and only if $\sup\{\Phi_0(b):1\geq b\in\Sim(\A_0)\}<\infty$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma semi-lattice} one may write (\textit{i})--(\textit{iii}) with $\Phi$ and $\A$
in place of $\Phi_0$ and $\A_0$, respectively. An additive set function on a ring is positive or
bounded if and only if its integral over simple functions is so, see e.g.
\cite[theorems 3.1.9, p. 63 and 3.2.5 p. 65]{rao}.
\end{proof}

Of some interest is also a version of Theorem \ref{th extension} for product spaces. If
$\Phi:\AB{}\to Y$, $A\in\A$ and $B\in\mathscr B$ then we define $\Phi^A:\mathscr B\to Y$
and $\Phi^B:\A\to Y$ by letting
\begin{equation*}
\Phi^A(B)=\Phi^B(A)=\Phi(A\times B)\qquad A\in\A,\ B\in\mathscr B
\end{equation*}
$\Phi$ is said to be separately in a given class if $\Phi^B$ and $\Phi^A$ belong to that class
for each $A\in\A$ and $B\in\mathscr B$.

\begin{lemma}
\label{lemma fubini}
Let $\A_0$ and $\mathscr B_0$ be semi-lattices of sets generating the lattices $\A_1$ and
$\mathscr B_1$ and the rings $\A$ and $\mathscr B$, respectively. Let also $\Phi_0:\AB{0}\to Y$.
Then:
\begin{enumerate}[(i)]
\item $\Phi_0$ is separately semi-modular if and only if there exists a unique separately
modular set function $\Phi_1:\AB{1}\to Y$ such that $\restr{\Phi_1}{\AB{0}}=\Phi_0$;
\item $\Phi_0$ is separately semi-additive if and only if there exists a unique separately
additive set function $\Phi:\AB{}\to Y$ such that $\restr{\Phi}{\AB{0}}=\Phi_0$.
\end{enumerate}
\end{lemma}

\begin{proof}
We prove the claim just for the case in which $\A_0$ is a $\cup$-lattice and $\mathscr B$ a
$\cap$-lattice but it will be clear that the argument remains unchanged for any other choice.
For each $B\in\mathscr B$ extend $\Phi^B$ to a semi-modular set function on $\A_1$ and define
the set function $\Phi_*:\A_1\times\mathscr B\to Y$ obtained from this by letting $B$ vary across
$\mathscr B$. We claim that $\Phi_*$ is separately semi-modular. In fact, let
$B,B_1,\ldots,B_N\in\mathscr B_0$ be such that $B=\bigcup_{n=1}^NB_n$ and fix $A\in\A_1$.
By Lemma \ref{lemma semi-lattice} there exist $A_1,\ldots,A_M\in\A_0$ such that
$A=\bigcap_{m=1}^MA_n$. Then,
\begin{align*}
\Phi_*(A\times B)&=\sum_{\emp<b\le [M]}\nu(b)\Phi_0\left(\bigcup_{m\in b}A_m\times B\right)\\
&=\sum_{\emp<a\le [N]}\nu(a)\sum_{\emp<b\le [M]}\nu(b)
\Phi_0\left(\bigcup_{m\in b}A_m\times\bigcap_{n\in a} B_n\right)\\
&=\sum_{\emp<a\le [N]}\nu(a)\Phi_0\left(A\times\bigcap_{n\in a} B_n\right)\\
&=\sum_{\emp<a\le [N]}\nu(a)\Phi_*\left(A\times\bigcap_{n\in a} B_n\right)
\end{align*}
so that $\Phi_*$ is semi-modular in its second coordinate while modular in the first one, by
construction; moreover it is unique with these properties. A further application of this same
argument proves the existence claim in (\textit{i}). If $\Psi_1:\AB{1}\to Y$ has the same
properties of $\Phi_1$ in (\textit{i}), then necessarily
$\restr{\Psi_1}{\A_1\times\mathscr B}=\restr{\Phi_1}{\A_1\times\mathscr B}$ and,
by semi-modularity in the second coordinate, $\Psi_1=\Phi_1$. The same argument applies in the
proof of the existence in (\textit{ii}). For fixed $B\in\mathscr B_1$, consider the set function
on $\A$ associated with $\Phi_1^B$ via \eqref{pettis} and obtain from this a set function
$\Phi_{**}:\A\times\mathscr B_1\to Y$. As above, the linear nature of the relationship linking
$\Phi_{**}$ to $\Phi_1$ implies that $\Phi^A_{**}:\mathscr B_1\to Y$ is modular. A further
application of this same procedure concludes the proof of existence in (\textit{ii}). As
above, uniqueness follows from Theorem \ref{th pettis} applied coordinatewise.
\end{proof}

\begin{theorem}
\label{th fubini}
Let $\A_0$, $\mathscr B_0$ be semi-lattices and $\mathscr C$ the ring generated by $\AB{0}$.
$\Phi_0:\AB{0}\to Y$ is separately semi-additive if and only if there exists
$\Phi\in sa(\mathscr C,Y)$ such that $\restr{\Phi}{\AB{0}}=\Phi_0$.
\end{theorem}

\begin{proof}
Define the set function $\Phi_*:(\A_0\cup\emp)\times(\mathscr B_0\cup\emp)\to Y$ by letting
$\Phi_*(A\times B)=\0$ if either $A=\emp$ or $B=\emp$ and
$\Phi_*(A\times B)=\Phi_1(A\times B)$ if $A\times B\in\AB{0}$. Given that $\Phi_1$ is semi-
additive on both coordinates, this is easily seen to be the unique separately semi-additive
extension of $\Phi_1$ to $(\A_0\cup\emp)\times(\mathscr B_0\cup\emp)$. To avoid additional
notation we simply assume $\emp\in\A_0$ and $\emp\in\mathscr B_0$ and $\Phi_0(\emp)=\0$.
Let $\A$ and $\mathscr B$ be the rings generated by $\A_0$ and $\mathscr B_0$ respectively.
By Lemma \ref{lemma fubini} we obtain a unique extension $\Phi$ to $\AB{}$ that is separately
additive. Let $\C_0$ be the collection of all finite unions $\bigcup_{n=1}^NA_n\times B_n$
from $\AB{}$ where the sets $A_1,\ldots,A_N$ are pairwise disjoint. Let
$C,\bar C\in\C_0$ be given, with $C=\bigcup_{j=1}^JA_j\times B_j$ and
$\bar C=\bigcup_{k=1}^K\bar A_k\times\bar B_k$. Writing
\begin{equation}
\label{union}
C\cup\bar C=\bigcup_{j,k}[(A_j\cap\bar A_k)\times(B_j\cup\bar B_k)]
\cup\bigcup_{j=1}^J\left[\left(A_j\backslash\bigcup_{k=1}^K\bar A_k\right)\times B_j\right]
\cup\bigcup_{k=1}^K\left[\left(\bar A_k\backslash\bigcup_{j=1}^JA_j\right)\times\bar B_k\right]
\end{equation}
shows that $\C_0$ is a $\cup$-lattice. If $C=\bar C$, then $A_j\cap\bar A_k\neq\emp$
implies $B_j=\bar B_k$ so that, $ \sum_{j=1}^J\Phi(A_j\times B_j)
=\sum_{j=1}^J\sum_{k=1}^K\Phi((A_j\cap\bar A_k)\times B_j)
=\sum_{k=1}^K\sum_{j=1}^J\Phi((A_j\cap\bar A_k)\times\bar B_k)
=\sum_{k=1}^K\Phi(\bar A_k\times\bar B_k) $. $\Phi_*:\C_0\to Y$ may thus be
unambiguously defined by letting
\begin{equation}
\Phi_*(C)=\sum_{j=1}^J\Phi(A_j\times B_j) \qquad
C=\bigcup_{j=1}^JA_j\times B_j\in\C_0
\end{equation}

To show that $\Phi_*\in sa_0(\C_0,Y)$, fix $C_1,\ldots, C_J,\bigcap_{j=1}^JC_j\in\C_0$ with
$C_j=\bigcup_{k=1}^{K_j} A^j_k\times B^j_k$. Given that $\A$ is a ring one may form a collection
$A_1,\ldots,A_N\in\A$ such that $A_k^j=\bigcup_{\{n:A_n\subset A_k^j\}}A_n$ for all $j,k$. If
$1\le n\le N$ and $1\le j\le J$ there is then at most one integer $k_n^j$ such that
$A_n\subset A_{k_n^j}^j$ otherwise let $A_{k_n^j}^j=\emp$.
Then,
\begin{align*}
\sum_{\emp<b\le[J]}\nu(b)\Phi_*\left(\bigcup_{j\in b}C_j\right)
&=\sum_{\emp<b\le[J]}\nu(b)\Phi_*\left(\bigcup_{n=1}^N\bigcup_{j\in b}A_n
\times B^j_{k_n^j}\right)\\
&=\sum_{\emp<b\le[J]}\nu(b)\Phi_*\left(\bigcup_{n=1}^NA_n\times
\left(\bigcup_{j\in b} B^j_{k_n^j}\right)\right)\\
&=\sum_{n=1}^N\sum_{\emp<b\le[J]}\nu(b)\Phi\left(A_n\times
\left(\bigcup_{j\in b} B^j_{k_n^j}\right)\right)\\
&=\sum_{n=1}^N\Phi\left(A_n\times\left(\bigcap_{j=1}^J B^j_{k_n^j}\right)\right)\\
&=\Phi_*\left(\bigcup_{n=1}^NA_n\times\left(\bigcap_{j=1}^J B^j_{k_n^j}\right)\right)\\
&=\Phi_*\left(\bigcap_{j=1}^JC_j\right)
\end{align*}
i.e. $\Phi_*\in sa_0(\C_0,Y)$. The claim thus follows from Theorem \ref{th pettis}.
\end{proof}
One may remark that Theorem \ref{th fubini} is somehow more general than Theorem
\ref{th extension} since the given initial collection, $\AB{0}$, is not even a semi-lattice nor is the
set function $\Phi_0$ semi-modular. Of course, Theorem \ref{th fubini} may be proved for any
arbitrary but finite number of coordinates.

\section{Order Semilattices}
Viewing the semi-lattice property considered above as a characteristic of the partial order induced
by set inclusion induces to extend this notion to partially ordered sets. A set $X$
endowed with a partial order $\ge$ is a $\wedge$-lattice (resp. $\vee$-lattice) where each
subset $\{x,y\}$ of $X$ admits an infimum $x\wedge y$ (resp. supremum $x\vee y$) in $X$.
This section proves a general version of a result originally due to Norberg \cite{norberg} by
which vector-valued functions on $X$ may be associated to semiadditive set functions on
special families of subsets of $X$. This fact may be regarded as a further example of this
family of set functions, adding to the ones of section 2. But, more importantly,
it shows, similarly to the Stiltijes construction, how to associate additive set functions
to point functions, even if of unbounded variation.

As usual, $\ge$ decomposes into an asymmetric part, $>$, and a symmetric part $\sim$,
defined implicitly by writing $x>x'$ whenever $x\ge x'$ but not $x'\ge x$ and $x\sim x'$
whenever $x\ge x'$ and $x'\ge x$. On the basis of these orderings, define the order intervals
\begin{subequations}
\label{segments}
\begin{align}
\label{segmentso}
\rc{x}&=\{x'\in X:x'\leq x\}&\lc{x}&=\{x'\in X:x'\geq x\}\\
\label{segmentsp}
\ro{x}&=\{x'\in X:x'<x\}&\lo{x}&=\{x'\in X:x'>x\}
\end{align}
\end{subequations}
A special property is the following:
\begin{equation}
\label{order}
\lo x\subset\lo{x'}\text{ implies }x\ge x'
\end{equation}
Indeed property \eqref{order} is rather natural and and is clearly true if $X$ is a space of
real-valued functions ordered pointwise. Another example requires some form of continuity.

\begin{example}
Let $\ge$ be upper semicontinuous in the sense that $x$ is a point of closure of $\lo x$
and that $\lc x$ is closed. Then \eqref{order} holds. If fact, if $\lo x\subset\lo{x'}$ then
$x\in\overline{\lo x}\subset\overline{\lo{x'}}\subset\lc{x'}$.
\end{example}

 From our point of view, \eqref{order} has an important implication.

\begin{lemma}
\label{lemma order}
Let $X$ be a $\wedge$-lattice possessing property \eqref{order} and let
$x_1,\ldots,x_N,y_1,\ldots,y_K\in X$. Then
$\bigcup_{n=1}^N\lo{x_n}\subset\bigcup_{k=1}^K\lo{y_k}$
implies $\lo{\bigwedge_{n=1}^Nx_n}\subset\lo{\bigwedge_{k=1}^Ky_k}$
or, equivalently, $\bigwedge_{n=1}^Nx_n\ge\bigwedge_{k=1}^Ky_k$.
\end{lemma}
\begin{proof}
$\lo{x_n}\subset\bigcup_{k=1}^K\lo{y_k}\subset\lo{\bigwedge_{k=1}^Ky_k}$and
\eqref{order} imply $x_n\ge\bigwedge_{k=1}^Ky_k$ for $n=1,\ldots,N$ or, equivalently,
$\bigwedge_{n=1}^Nx_n\ge\bigwedge_{k=1}^Ky_k$ which, under \eqref{order}, is the
same as $\lo{\bigwedge_{n=1}^Nx_n}\subset\lo{\bigwedge_{k=1}^Ky_k}$.
\end{proof}

Define also the following families of subsets of $X$
\begin{align}
\label{AB}
\A_0&=\left\{\rc x:x\in X\right\}&
\mathscr B_0&=\left\{\bigcup_{n=1}^N\lo{x_n}:N\in\N,\ x_1,\ldots,x_N\in X\right\}
\end{align}

If $Y$ is a real vector space, then so is
$\mathfrak F(X,Y)=\left\{F\in Y^{X}:x\sim x'\text{ implies }F(x)=F(x')\right\}$.

\begin{corollary}[Norberg]
\label{corollary norberg}
Let $X$ be a $\wedge$-lattice. Then $\mathfrak F(X,Y)$ is linearly isomorphic to $sa_0(\A_0,Y)$
via the identity
\begin{align}
\label{bijection}
\Phi(\rc{x})&=F(x)&x\in X
\end{align}
\end{corollary}

\begin{proof}
Observe that $\A_0$ is a $\cap$-lattice and $\emp\notin\A_0$. Given that
$F\in\mathfrak F(X,Y)$ and that $\rc{x}=\rc{x'}$ if and only if $x\sim x'$, writing the right
hand side of \eqref{bijection} as $\Phi(\rc{x})$ implicitly defines a set function
$\Phi:\A_0\to Y$. Assume that $\{x_1,\ldots,x_N\}\subset X$ is such that
$\bigcup_{n=1}^N\rc{x_n}\in\A_0$ i.e. such that $\bigcup_{n=1}^N\rc{x_n}=\rc{x}$ for
some $x\in X_0$. But then, upon relabeling, $x_1,\ldots,x_N\le x_0\le x_1$ so that
$\bigcup_{n=1}^N\rc{x_n}=\rc{x_1}$. Thus, $\Phi\in sa_0(\A_0,Y)$, by Lemma
\ref{lemma pre norberg}.
\end{proof}

Corollary \ref{corollary norberg} may be stated for $\vee$-lattices after replacing
$\rc x$ with $\lc x$ in the definition of $\A_0$. The proof is virtually
unchanged.

This result was first stated by Norberg in a noteworthy but unpublished paper, \cite[pp. 6-9]{norberg}.
His approach was later revived by the literature on set-indexed stochastic processes, although in
a rather troublesome way (see section 5)

A strictly related but more difficult question is whether it is possible to associate
$F\in\mathfrak F(X,Y)$ with a set function on the family of strict order intervals \eqref{segmentsp}.
The novelty here is that this class of intervals is not itself a semi-lattice and the additional condition
\eqref{order} is therefore required.

\begin{theorem}
\label{th norberg}
Let $X$ be a $\wedge$-lattice and \eqref{order} hold. Then $\mathfrak F(X,Y)$ is
linearly isomorphic to $sa_0(\mathscr B_0,Y)$ via the identity
\begin{align}
\label{bijection>}
\Psi\left(\bigcup_{n=1}^N\lo{x_n}\right)&=F\left(\bigwedge_{n=1}^Nx_n\right)+y_F
& x_1,\ldots,x_n\in X
\end{align}
where $y_F\in Y$ is unique (given $F$).
\end{theorem}

\begin{proof}
$\mathscr B_0$ is clearly a $\cup$-lattice. By Lemma\ref{lemma order} , the quantity
$F\left(\bigwedge_{n=1}^Nx_n\right)$ induces a well defined set function $\Psi_0$ on
$\mathscr B_0$ with the property that
$\Psi_0(\bigcup_{n=1}^N\lo{x_n})=\Psi_0(\lo{\bigwedge_{n=1}^Nx_n})$. In order
to prove that $\Psi_0$ is semi-modular, assume that $x(n,m),y_k\in X$ for $m=1,\ldots,M_n$,
$n=1,\ldots,N-1$ and $k=1,\ldots,K$ are such that
\begin{equation}
\label{=}
\bigcap_{n=1}^{N-1}\bigcup_{m=1}^{M_n}\lo{x(n,m)}=\bigcup_{k=1}^K\lo{y_k}
\end{equation}
Let for simplicity $x_n=\bigwedge_mx(n,m)$, $y=\bigwedge_ky_k$, $A_n=\lo{x_n}$
for $n=1,\ldots,N-1$ and $A_N=\lo y$. It is easily deduced from \eqref{=} that
$A_N\subset\bigcap_{n=1}^{N-1}A_n$ so that
$\Psi_0\left(\bigcup_{n\in b}A_n\right)
=\Psi_0\left(A_N\cup\bigcup_{n\in b}A_n\right)$ for all $\emp<b\le[N-1]$. But then,
\begin{align*}
\sum_{\emp<n\le[N-1]}\nu(b)\Psi_0\left(\bigcup_{n\in b}A_n\right)
&=\sum_{\emp<n\le[N]}\nu(b)\Psi_0\left(\bigcup_{n\in b}A_n\right)&\text{ by }\eqref{parsimony}\\
&=\Psi_0(A_N)&\text{ by }\eqref{decomposition}\\
&=\Psi_0\left(\bigcup_{k=1}^K\lo{y_k}\right)&\text{ by }\eqref{bijection>}\\
&=\Psi_0\left(\bigcap_{n=1}^{N-1}\bigcup_{m=1}^{M_n}\lo{x(n,m)}\right)&\text{ by }\eqref{=}
\end{align*}
The claim then follows from Corollary \ref{corollary semiadditive}.
\end{proof}

Given its importance, in the following sections when dealing with a $\wedge$-lattice
we will always assume that property \eqref{order} is satisfied.

\section{Stochastic Processes Indexed by a semi-lattice}
In this section we provide applications of the preceding measure-theoretic results to stochastic
process when the index set $T$ is a semi-lattice. The main advantage of this approach, as
compared e.g. to the literature on set-indexed processes, lies in the limited number of
assumptions required. We only need to fix a set $\Omega$, a semi-lattice $T$ and a collection
$\FFT$ of $\sigma$ algebras of subsets of $\Omega$ such that $s,t\in T$ and $s\leq t$ imply
$\F_s\subset\F_t$; we also posit the existence of a (countably additive) probability $P$ on
$\F=\bigvee_{t\in T}\F_t$.
We write $L^1$ for $L^1(\Omega,\F,P)$ and $\norm{\cdot}$ for $\norm\cdot_{L^1}$. $\bar\Pred$
and $\bar\Opt$ will denote the rings generated by the collections
\begin{align*}
\left\{F\times\lo{t}:F\in\F, t\in T\right\} \quad\text{and}\quad
\{F\times\lc{t}:F\in\F, t\in T\}
\end{align*}
respectively (see \eqref{segments} for notation). We shall also refer to the collections
\begin{equation}
\label{T0}
\T_0^p=\left\{\bigcup_{n=1}^N\lo{t_n}:N\in\N,\ t_1,\ldots,t_N\in T\right\}
\quad\text{and}\quad\T_0^o=\left\{\lc t:t\in T\right\}
\end{equation}
and to the generated rings, $\T^p$ and $\T^o$ respectively.

A similar setting was considered in \cite{SPTA}, where the index set is a family of subsets of
$\Omega\times I$ with $I$ a partially ordered set, and can thus be interpreted as a collection
of stochastic intervals. The results in \cite{SPTA} can then be regarded as describing the
properties of processes indexed by stopping times rather than deterministic times, as is the
case here. If the underlying index set is not linearly ordered, this difference is rather deep as
illustrated, e.g., by the failure of the optional sampling theorem, see \cite{hurzeler}.

We shall occasionally exemplify our results on the basis of the following setting:
\begin{example}[Experiment design]
\label{ex experiment}
Let $(X_k(n):n\in\R_+)$ describe the random outcome of some experiment in a given location $k$,
with $X_k(0)=0$. Designing an experiment is a matter of choosing which locations will participate
and at what time (time $0$ corresponds to non participation). An experiment design may thus be
viewed as a sequence $\sseq{t(k)}k$ that assigns at each location $k$ the exact time $t(k)$ at
which the experiment will have to take place. Given that only a limited number of locations may
be involved, the set $T$ of all designs consists of sequences with only a finite number of non null
terms. $T$ is endowed with pointwise order so that if $t,u\in T$ and $u\ge t$ then $u$ involves
a larger number of locations, each being active at a later moment in time  than $t$. This partial
order reflects the view that the larger the number of units involved and the later the moment of
the experiment the more informative its outcome will be. A special case occurs when the
experiment may only involve one single location so that the corresponding index set
$T_0\subset T$ consists of sequences with only one non null element. It is clear that $T_0$ is
isomorphic to $\R^2$ as each sequence $t\in T_0$ such that $t(n)=0$ whenever $n\ne k$ may
actually be represented as the pair $(k,t(k))$. The aggregate outcome of the experiment will
be some statistics of the sample $(X_k(t(k)):k\in\N)$, e.g. the sample mean or the maximum
\begin{equation}
\label{mean}
X^m_t=\frac{1}{\#\{k\in\N:t(k)>0\}}\sum_kX_k(t(k))\qquad
X^*_t=\max_kX_k(t(k))
\end{equation}
A natural development would be to consider the possibility of random experimental schedules, that
is of functions $\kappa:S\to T$ where $(S,\Sigma)$ is some given measurable space. $\kappa$
would then play the role of a generalized stopping time indicating the random time and the
random place where the experiment will take place.
\end{example}

\subsection{Stochastic Processes and Induced Set Functions.}

We denote the class of generalized stochastic processes by
\begin{equation}
\label{Xi}
\Xi=\left\{Y\in\underset{t\in T}\times L^0:P(Y_s=Y_t)=1\text{ whenever }s\sim t\right\}
\end{equation}
and say that $Y\in\Xi$ is locally integrable if $Y_t\in L^1$ for all $t\in T$ or integrable if
$\sup_{t\in T}P(\abs{Y_t})<\infty$. A stochastic process is an element $X\in\Xi$ which is
adapted, i.e. such that $X_t$ is $\F_t$ measurable for all $t\in T$. Given the possibility of
adding a point at infinity, we assume with no loss of generality that $T$ admits a largest
element $t_\infty$ and write $X_\infty=X_{t_\infty}$ or, if $t_\infty$ is a point added at
infinity, $X_\infty=0$\footnote{The existence of a largest element $t_\infty$ implies
$\emp\in\T_0^p$; while $\emp\notin\T_0^o$.}.
Looking just at ordered pairs $s,t\in T$, $s\le t$, we can define in the present setting most
of the classical properties and say, e.g., that $X$ is a supermartingale
(resp. an increasing process) if $X_s\ge\condP{X_t}{\F_s}$ (resp. $P(A_s\le X_t)=1$) for
all such pairs. Given the extension procedure to be described shortly, it should be stressed
that these properties depend on the filtration considered and on the index set adopted.

The next result associates any $X\in\Xi$ with a set function on $\bar\Pred$.

\begin{theorem}
\label{th process}
Let $T$ be a semi-lattice. $\Xi$ is linearly isomorphic to either $sa(\bar\Pred,L^0)$ or
$sa(\bar\Opt,L^0)$) via the following identities holding for all  $t_1,\ldots,t_N\in T$ and
$F\in\F$:
\begin{subequations}
\label{Phi}
\begin{align}
\label{Phip}
\bar\Phi^p_X\left(F\times\bigcup_{n=1}^N\lo{t_n}\right)&=
\set{F}\left(X_\infty-X_{\bigwedge_{n=1}^Nt_n}\right)
\end{align}
\begin{align}
\label{Phio}
\bar\Phi^o_X\left(F\times\bigcap_{n=1}^N\lc{t_n}\right)&=
\set{F}\left(X_\infty-X_{\bigvee_{n=1}^Nt_n}\right)
\end{align}
\end{subequations}

\end{theorem}

\begin{proof}
Both $\T_0^p$ and $\T_0^o$ are semilattices under the corresponding assumption on $T$,
as $\bigcap_{n=1}^N\lc{t_n}=\dlc{\bigvee_{n=1}^Nt_n}$. Moreover,
$\bar\Pred\subset\F\times\T^p$ and $\bar\Opt\subset\F\times\T^o$. In order to apply
Theorem \ref{th fubini} we need to show that the right hand sides of \eqref{Phi} describe
well defined set functions on $\F\times\T^p_0$ and $\F\times\T^o_0$ respectively. In fact
semi-additivity is clear, in view of Theorem \ref{th norberg}. Observe then that
$F\times\lc{s}=G\times\lc{t}$ implies $F=G$ and either $F=\emp$ or $s\sim t$:
thus $\set{F}(X_\infty-X_s)=\set{G}(X_\infty-X_t)$. On the other hand,
$F\times\bigcup_{n=1}^N\lo{t_n}=\emp$ only if either $F=\emp$ or
$\bigcup_{n=1}^N\lo{t_n}=\lo{t_\infty}$ from which, given \eqref{order}, follows that
$\bigwedge_{n=1}^Nt_n=t_\infty$. In either case,
$\set F(X_\infty-X_{\bigwedge_{n=1}^Nt_n})=\emp$.
Moreover, $F\times\bigcup_{n=1}^N\lo{s_n}=G\times\bigcup_{k=1}^K\lo{t_k}\ne\emp$ implies
$F=G$ and $\bigcup_{n=1}^N\lo{s_n}=\bigcup_{k=1}^K\lo{t_k}$ and, as shown in the
proof of Theorem \ref{th norberg}, $\bigwedge_{n=1}^Ns_n\sim\bigwedge_{k=1}^K t_k$
so that indeed the right hand side of \eqref{Phip} defines a set function on $\F\times\T_0^p$.
\end{proof}

The relationship between processes and measures established in Theorem \ref{th process} is
a first step of well known importance in the construction of the stochastic integral proposed by
\^{I}to as well as in the Dol\'{e}ans-Dade approach to the Doob Meyer decomposition. Our
choice to define such measure on $\bar\Pred$ (resp. $\bar\Opt$) is somehow unconventional
as it disregards the issue of adaptness but is motivated by the need to exploit the semi-lattice
property.

Let us assume for the rest of the paper that $T$ is a $\wedge$-lattice (the $\vee$ case may
be treated similarly). The first use of Theorem \ref{th process} is to obtain an extension of
the original index set to the $\T^p$. If we  identify $t\in T$ with $\lo{t}\subset\T^p$ then
$\T^p$ is easily seen to be an extension of $T$; moreover, $\T^p$ inherits the order from
$T$ if we let $\tau\le\upsilon$ whenever $\upsilon\subset\tau$: $\emp$ is thus the unique
maximal element. Likewise, one may fix a filtration $(\F_\tau:\tau\in\T^p)$ with
$\F_t\subset\F_\tau$ whenever $\tau\subset\lo{t}$ and $\F_\emp=\F$. For example,
for $\sigma\in\T_0^p$, $\tau\in\T_1^p$ and $\upsilon\in\T^p$ one may set
\begin{equation}
\label{filtration}
\F^0_\sigma=\bigvee_{\left\{\sigma=\bigcup_{n=1}^N\lo
{t_n}\right\}}\bigcap_{n=1}^N\F_{t_n},
\quad
\F^1_\tau=\bigcap_{\{\sigma\in\T_0^p:\tau\subset\sigma\}}\F^0_\sigma
\quad\text{and}\quad
\F_\upsilon=\bigcap_{\{\tau_1,\tau_2\in\T_1^p:\tau_1\cap\tau_2^c\subset\upsilon\}}\F^1_{\tau_2}
\end{equation}

In analogy with \eqref{Xi}, we designate the class of generalized stochastic processes on this
new index set by $\bar\Xi$. More importantly, we define the ring of predictable sets
\begin{equation}
\label{Pred}
\Pred=\{F\times\tau:F\in\F_\tau,\ \tau\in\T^p\}
\end{equation}
and the restriction $\Phi_X^p=\restr{\bar\Phi_X^p}{\Pred}$. Our definition of $\Pred$ compares
somehow to that proposed by Saada and Slonowsky \cite[p. 70]{saada slonowsky}\footnote{
In \cite{saada slonowsky}, however, in place of $\T^p$ appears the (semi) ring of \textit{all}
subsets of the indexing class. The authors show its utility in the construction of the stochastic
integral.}

The notion of predictability implicit in \eqref{Pred} is delicate as several alternative definitions
are in fact possible. An example would be to let $\T^p$ be defined by the sets of the form
$\rc t^c$. In this case the intuitive meaning of \textit{prior to}, naturally associated with the
concept of prediction, would in fact coincide with the notion of \textit{no later than}. The nice
fact is that Theorem \ref{th process} above would carry through unchanged to this case as
to all other definitions that allow to view each $X\in\Xi$ as defining a set function on a
semi-lattice of subsets of $\Omega\times T$. Observe that in the classical case, $T=\R_+$,
predictability may be equivalently defined in terms of order (i.e. starting from left-open stochastic
intervals) or of topology (i.e. starting with the class of \caglad processes)}. Another possible line of
approach, taken, e.g., by the literature on set-indexed processes, suggests to start with a
convenient set of topological assumptions on $T$\footnote{See for example \cite{ivanoff merzbach}
where the indexing family consists of compact subsets of a $\sigma$-compact space and is in
fact closed under arbitrary intersections. More or less the same setting appears in all contributions
to this literature}. The difference with this way of thinking of predictability is indeed rather deep
when $T$ is just partially ordered and makes the comparison with the definition given here
hardly possible.

Extending the process $X$ to this enriched setting may now be done quite simply.
\begin{corollary}
\label{cor process}
For each $X\in\Xi$ there exists a unique process $\bar X=(\bar X_\tau:\tau\in\T^p)\in\bar\Xi$
such that
\begin{equation}
\label{X extension}
\bar X_\tau=X_{\bigwedge_{n=1}^Nt_n}
\qquad t_1.\ldots,t_N\in T,\ \tau=\bigcup_{n=1}^N\lo{t_n}
\end{equation}
and that
\begin{equation}
\label{X additive}
\bar X_{\bigcup_{n=1}^N\tau_n}=\sum_{\emp<b\le[N]}\nu(b)\bar X_{\bigcap_{n\in b}\tau_n}
\quad\text{and}\quad
\bar X_{\bigcap_{n=1}^N\tau_n}=\sum_{\emp<b\le[N]}\nu(b)\bar X_{\bigcup_{n\in b}\tau_n}
\quad t_1,\ldots,t_N\in\T^p
\end{equation}
Moreover, $\bar X$ is adapted to the filtration defined in \eqref{filtration}.
\end{corollary}

\begin{proof}
It is enough to define:
\begin{equation}
\label{X_tau}
\bar X_\tau=X_\infty-\Phi^p_X(\Omega\times\tau)\qquad\tau\in\T^p
\end{equation}
Then \eqref{X extension} follows from \eqref{Phi} while \eqref{X additive} is an immediate
consequence of the fact that $\Phi_X^p$ is additive. If $\bar Z\in\bar\Xi$ were another such extension, then,
by \eqref{X additive}, the set function on $\bar\Pred$ defined implicitly by letting
$\lambda(F\times\tau)=\set F(X_\infty-Z_\tau)$ for all $F\in\F$ and $\tau\in\T^p$ would
be additive. Moreover, $\lambda$ coincides with $\Phi_X^p$ in restriction to the semi-lattice
$\F\times\T_0^p$. But we then conclude from Theorem \ref{th extension} that necessarily
$\lambda=\bar\Phi_X^p$ and thus $\bar Y_\tau=\bar X_\tau$ for all $\tau\in\T^p$, by
\eqref{X_tau}. That $\bar X$ is adapted to $(\F_\tau:\tau\in\T^p)$ follows from Lemma
\ref{lemma semi-lattice} and \eqref{filtration}.
\end{proof}

The claim that X extends additively to $(\F_\tau:\tau\in\T^p)$, proved in Corollary
\ref{cor process}, had already appeared, in a slightly different context, in \cite{dozzi} and in
several papers that followed that work, but unfortunately the argument presented was
erroneous. This property has then later been taken as a key assumption in this literature
(see \cite{ivanoff merzbach} for details and \cite{SPTA} for remarks) while it follows here
quite naturally from Theorem \ref{th extension}.

Given the better properties of rings versus semilattices, it is quite natural to prefer working with
its extension rather than with the original process. However, one should be warned that the former
may not possess some of the original mathematical properties, such as the supermartingale
property\footnote{The literature on set-indexed processes, started from Cairoli and Walsh
\cite[Definition p. 115]{cairoli walsh}, makes use of the terms \textit{weak} and \textit{strong}
to distinguish between properties possessed by $(X_t:t\in T)$ rather than by its externsion
$(\bar X_\tau:\tau\in\T)$.}.

To conclude on the role of $\Phi_X^p$, let us mention that the construction of the stochastic
integral consists mainly of the extension of the map $\int hdX=\Phi_X^p(h):\Sim(\Pred)\to L^0$
to more general spaces. Also relevant, when $X$ is locally integrable, is the possibility to define
the Dol\'{e}ans-Dade measure associated with $X$, namely
\begin{align}
\label{phi}
\phi^p_X&=P\otimes\Phi_X^p
\end{align}

We develop in the following subsections some implications of the preceding results aiming
in particular at replicating some classical decompositions. All processes encountered
will be element of $\bar\Xi$.

\subsection{Quasi Martingales}
A major role is played by the family $\D$ of all finite, disjoint collections $d$ of elements of
$\T^p$ of the form $\tau\cap\upsilon^c$ with $\upsilon\subset\tau$. Since $\T^p$ is a ring,
the refinement relation makes $\mathscr D$ into a directed set.

If $X$ is a locally integrable stochastic process one can define
\begin{equation}
\label{d variation}
V^d(X)=\Vd{X}\quad S^d(X)=\Sd{X}\qquad d\in\D
\end{equation}
and $\mathcal S(X)=\{S^d(X):d\in\D\}$. We refer to $X$ as a quasi martingale, in symbols $X\in\QQ$,
whenever $\norm{X}_\QQ:=\sup_{d\in\D}\dnorm{V^d(X)}<\infty$\footnote{In the literature quasi-
martingales are also called $F$-processes or processes of bounded variation.}. Differently from the
classical case, supermartingales, submartingales or even increasing processes need not be quasi
martingales. We provide an example.

\begin{example}
Resume the setting of Example \ref{ex experiment} and let $\seq{A}{k}$ be a sequence of one
dimensional increasing processes (thus $A_k(0)=0$) bounded by $1$ describing the local experiment.
Let also $A^*$ be the sample maximum \eqref{mean} of all local experiments, itself an increasing
process bounded by $1$. Consider the segment $]a,a+1]\subset\R_+$ and a subset $K\subset\N$
and define $\tau_K(a)\in T$ by letting $\tau_K(n)=0$ if $n\notin K$ and $\tau_K(n)=a$ otherwise.
We can then define $\tau_K(]a,a+1])=\{t\in T: \tau_K(a)<t\le\tau_K(a)\}\in\T^p$. It is clear that
$K\cap K'=\emp$ implies $\tau_K(]a,a+1])\cap\tau_{K'}(]a,a+1])=\emp$ so that we can easily
form a disjoint countable collection $\{\tau_{K_i}(]a,a+1]):i\in\N\}$ . But then
\begin{align*}
\norm{A^*}_\QQ&\ge
P\sum_i\dabs{\condP{A^*(\tau_{K_i}(a+1))}{\F_{\tau_{K_i}(a)}}-A^*(\tau_{K_i}(a))}\\
&=P\sum_i\max_{k\in K_i}\dabs{\condP{A_k(a+1)}{\F_{\tau_k(a)}}-A_k(a)}
\end{align*}
If, for example, $A_k(a+1)>\eta+ A_k(a)$ for all $k\in\N$ then clearly $A$ is not a quasi martingale.
This conclusion applies also to the special case in which the index set is $T_0$ and is therefore
isomorphic to $\R^2$.
\end{example}

Interest for quasi martingales originates from their role as integrators, following from the
isometry
$$
\dnorm{\Phi_X^p}=\sup_{\{h\in\Sim(\Pred):\norm{h}_\B\leq1\}}\dnorm{\Phi_X^p(h)}
=\norm{X}_\QQ
$$
Then, $X\in\QQ$ if and only if $\Phi_X^p:\Sim(\Pred)\to L^1$ is continuous when $\Sim(\Pred)$ is
endowed with the supremum norm. This provides a natural extension of the stochastic integral to
the closure of $\Sim(\Pred)$ in $\B$. Quasi martingales possess other, interesting properties.

If $\{\tau_\alpha\cap\upsilon^c_\alpha:\alpha\in\mathbf A\}$ is a disjoint collection
with $\tau_\alpha,\upsilon_\alpha\in\T$ and $\upsilon_\alpha\geq\tau_\alpha$, then,
\begin{equation}
\label{BV integrability}
\sum_{\alpha\in\mathbf A}\dnorm{\condP{X_{\upsilon_\alpha}}{\F_{\tau_\alpha}}-
X_{\tau_\alpha}}\le\dnorm{X}_{\QQ}
\end{equation}
This inequality, completeness of the space $L^1$ and contractiveness of conditional expectation
determine several interesting convergence results, as in the original work of Rao \cite{rao quasi}.
If, for example, one chooses the version
$\condP{X_{\upsilon_\alpha}}{\F_{\tau_\alpha}}=X_{\tau_\alpha}$ whenever
$\dnorm{\condP{X_{\upsilon_\alpha}}{\F_{\tau_\alpha}}-X_{\tau_\alpha}}=0$ then
\eqref{BV integrability} implies that the infinite sums
\begin{equation}
\label{VA}
S^{\mathbf A}(X)=
\sum_{\alpha\in\mathbf A} \condP{X_{\upsilon_\alpha}}{\F_{\tau_\alpha}}-X_{\tau_\alpha}
\quad\text{and}\quad
V^{\mathbf A}(X)=
\sum_{\alpha\in\mathbf A} \dabs{\condP{X_{\upsilon_\alpha}}{\F_{\tau_\alpha}}-X_{\tau_\alpha}}
\end{equation}
converge $P$ a.s. and in $L^1$. This conclusion applies, e.g., to monotone sequences
$\seqn{\tau}$ by setting $\upsilon_n=\tau_{n-1}\vee\tau_{n+1}$ and $\mathbf A=\N$.
Then, writing $\G_{\tau_k}=\bigcap_{j\geq k}\F_{\tau_j}$, one concludes that the process
$(Y_{\tau_k}:k\in\N)$ with
\begin{equation}
\label{Y}
Y_{\tau_k}=\condP{X_{\tau_j}}{\G_{\tau_k}}+\sum_{n\geq j}\condP{X_{\upsilon_n}-X_{\tau_n}}
{\G_{\tau_k}}\qquad j\geq k
\end{equation}
is integrable with $\norm{Y_{\tau_k}}\leq\norm{X_{\tau_k}}+\norm{X}_\QQ\leq2\norm{X}_\QQ$
and that
\begin{equation}
\label{convergence}
\lim_p\sup_k\dabs{\condP{X_{\tau_{k+p}}} {\G_{\tau_k}}-Y_{\tau_k}}\leq\lim_p\sum_ {j\geq p}
\dabs{\condP{X_{\upsilon_j}}{\F_{\tau_j}}-X_{\tau_j}}=0
\qquad P\text{ a.s. and in }L^1
\end{equation}
More interestingly,
\begin{lemma}
\label{lemma BV}
Let $\seqn{\tau}$ be an increasing sequence in $\T^p$ and $X\in\QQ$. Then, (i) $Y_{\tau_k}=
\lim_n\condP{X_{\tau_n}}{\F_{\tau_k}}$ exists $P$ a.s. and in $L^1$ and (ii)
$\lim_n\dnorm{Y_{\tau_n}-X_{\tau_n}}=0$. If $X$ is uniformly integrable then
$\sseqn{X_{\tau_n}}$ admits an $L^1$ limit which only depends on $\bigvee_n\tau_n$.
\end{lemma}

\begin{proof}
If $\seqn{\tau}$ is increasing, $\G_{\tau_k}=\F_{\tau_k}$. (\textit{i}) is just \eqref{convergence};
(\textit{ii}) easily follows from \eqref{Y}. By (\textit{ii}) and the fact  that $(Y_{\tau_k}:k\in\N)$ is
a martingale, $(X_{\tau_k}:k\in\N)$ converges in $L^1$ if and only if it is uniformly integrable.
Assume that $\seqn{\tau}$ and $\seqn{\bar\tau}$ are increasing and that
$\bigvee_n\bar\tau_n= \bigvee_n\tau_n$. Since each $\tau_n\in\T$ is the intersection of finitely
many intervals of the form $\lo{t_n^k}$ with $t_n^k\in T$, then, for each pair $n,k$ there must
be a pair $m,l$ such that $t_n^k\leq\bar t_m^l$, i.e. for each $n\in\N$ there is $m$ such that
$\tau_n\leq\bar\tau_m$. In yet other words, moving to subsequences if necessary and relabeling
terms, we may assume with no loss of generality that $\tau_n\leq\bar\tau_n\leq\tau_{n+1}$. But
then, considering the sequence $\sseqn{X_{\hat\tau_n}}$ obtained by letting
$\hat\tau_{2j-1}=\tau_j$ and $\hat\tau_{2j}=\bar\tau_j$ and in view of \eqref{convergence}
we conclude that necessarily $\lim_nX_{\tau_n}=\lim_nX_{\bar\tau_n}$.
\end{proof}

The classical Riesz decomposition (see \cite[theorem 1.1, p. 80]{rao}) easily follows.
\begin{corollary}
\label{corollary riesz}
$X\in\QQ$ is uniformly integrable if and only if it decomposes uniquely into the sum
\begin{equation}
\label{riesz}
X_\tau=\condP{M}{\F_\tau}+Z_\tau\qquad \tau\in\T^p
\end{equation}
where $M\in L^1$ and $Z\in\QQ$ is uniformly integrable with
$\lim_{\tau\in\T}\dnorm{X_\tau-M}=\lim_{\tau\in\T}\dnorm{Z_\tau}=0$. Moreover $Z$ in
\eqref{riesz} is in addition a positive supermartingale (i.e. a quasi potential) if and only if $X$
is a supermartingale.
\end{corollary}
\begin{proof}
Uniqueness is clear: if $M'$ and $Z'$ were another pair of processes satisfying \eqref{riesz},
then $\abs{Z_\tau-Z'_\tau}=\abs{M_\tau-M'_\tau}\leq\abs{M_\upsilon-M'_\upsilon}
=\abs{Z_\upsilon-Z'_\upsilon}$ so that $\norm{Z_\tau-Z'_\tau}=\norm{M_\tau-M'_\tau}=0$.
To prove existence, define the monotone set function $\lambda:\T\to\left[0,\norm{X_\QQ}\right]$
implicitly by letting
$$\lambda(\tau)=\sup_{d\in\D_\tau}P(V^d(X))\qquad\tau\in\T^p$$
and construct an increasing sequence $\seqn{\tau}$ in $\T^p$ such that $\lambda(\tau_n)
\le\inf_{\tau\in\T}
\lambda(\tau)+2^{-n}$. Let $M$ designate the $L^1$ limit of $\sseqn{X_{\tau_n}}$, which
exists by Lemma \ref{lemma BV},
and put $Z_\tau=X_\tau-\condP{M}{\F_\tau}$. If $\tau,\tau_n\in\T^p$, $\tau\geq\tau_n$,
$d_\tau\in\D_\tau$ and $d_n=\{\tau_n\cap\tau^c\}\cup d_\tau$, then $d_n\in\D_{\tau_n}$ and
$\dabs{X_\tau-X_{\tau_n}}=V^{d_n}(X)-V^{d_\tau}(X)$. It follows that
$\norm{X_\tau-X_{\tau_n}}\leq\lambda(\tau_n)-\lambda(\tau)\leq2^{-n}$ and, consequently, that
$\norm{Z_\tau}\leq\norm{X_\tau-X_{\tau_n}}+\norm{X_{\tau_n}-M}\leq2^{-n}+\norm{X_{\tau_n}
-M}$. $Z\in\QQ$ and is uniformly integrable since both $X$ and $M$ belong to this same class.
If $X$ is a supermartingale, then so is $Z$ and for each $F\in\F_\tau$,
$P(Z_\tau\set{F})\geq\lim_\upsilon P(Z_\upsilon\set{F}) =0$ so that $Z$ is positive and therefore
a quasi potential. The converse is clear.
\end{proof}

\subsection{Doob Meyer Decompositions}
We now consider general decompositions of locally integrable processes. In doing so we exploit
techniques similar to those developed in \cite{SPTA} but with the noteworthy difference that the
processes considered here do not necessarily generate bounded, positive set functions as was
for the case of class $D_0$ supermartingales treated in the mentioned reference.

For each $d\in\D$ define the operator $\Pred^d_P:L^\infty(P)\to\B(\Pred)$ as
\begin{equation}
\label{Pd}
\Pd(b)=\sum_{\tu\in d}\condP{b}{\F_\tau}\set{\tu}\qquad b\in L^\infty(P)
\end{equation}
Fix $F\in\F$ and $\tau\in\T$. If $d\geq\{\tau^c\}$ and $\bar\tau\cap\bar\upsilon^c\in d$ then
$\bar\tau\leq\bar\upsilon\leq\tau$ and thus
$\Pd(F)\set{\tau^c}=\Pd(\condP{F}{\F_\tau})\set{\tau^c}$.
Therefore
\begin{equation}
\label{mu-alpha}
\begin{split}
P(X_\tau\set{F})&=\phi_X^p(\set{\tau}\condP{F}{\F_\tau})\\
&=\phi_X^p(\set{\tau}\Pd(\condP{F}{\F_\tau}))\\
&=\phi_X^p(\Pd(\condP{F}{\F_\tau}))-\phi_X^p(\set{\tau^c}\Pd(\condP{F}{\F_\tau}))\\
&=\phi_X^p(\Pd(\condP{F}{\F_\tau}))-\phi_X^p(\set{\tau^c}\Pd(F))\\
&=\mu_X^d(\condP{F}{\F_\tau})-\alpha^d_{X,\tau}(F)
\end{split}
\end{equation}
The last line of \eqref{mu-alpha} defines $\mu^d_X,\alpha_{X,\tau}^d\in ca(\Omega,\F,P)$
implicitly as
\begin{align}
\label{phid}
\mu_X^d(F)&=\phi_X^p\left(\Pd(F)\right)=P\left(S^d(X)\set{F}\right)&
\alpha^d_{X,\tau}(F)&=\phi_X^p\left(\set{\tau^c}\Pd(F)\right)
&\tau\in \T^p,\ F\in\F
\end{align}

\begin{definition}
A stochastic process is said to possess the strong Doob Meyer property, or simply to be of class
$DM_s$, if the limit $\mu_X(F)=\lim_d\mu_X^d(F)$ exists for all $F\in\F$ and if $\mu_X\in ca(\F)$.
\end{definition}

Given that $\mu^d_{X+Y}=\mu^d_X+\mu^d_Y$ it is clear that if $X$ and $Y$ are of class $DM_s$
then so is $X+Y$. The following theorem motivates our terminology.

\begin{theorem}
\label{th DMs}
A stochastic process $X$ is of class $DM_s$ if and only if it admits the decomposition
\begin{equation}
\label{DM}
X_\tau=\condP{M}{\F_\tau}-A_\tau\qquad\tau\in\T^p,\ P\ a.s.
\end{equation}
where $M\in L^1$ and $A$ is a stochastic process of class $DM_s$ such that
\begin{equation}
\label{natural}
P(A_\tau\set{F})=\lim_dP\int_{\tau^c}\Pred^d(F)dA\qquad \tau\in\T^p,\ F\in\F
\end{equation}
Decomposition \eqref{DM}, if it exists, is unique. If $X$ is a supermartingale then $A$ is increasing.
\end{theorem}

\begin{proof}
It follows from \eqref{mu-alpha} that if the net $\nnet{\mu^d_X}{d}{\D}$ converges set wise to
some $\mu_X\in ca(\F)$ then so does the net $\nnet{\alpha^d_{X,\tau}}{d}{\D}$ for all
$\tau\in\T^p$. Let $\alpha_{X,\tau}$ be the corresponding limit and let $M,A_\tau\in L^1$ be
versions of the Radon Nikod\'{y}m derivatives of $\mu_X$ and $\alpha_{X,\tau}$, respectively.
\eqref{DM} soon follows. Moreover, if $\tau\in\T^p$ and $F\in\F$ then
\begin{align*}
P(A_\tau\set{F})&=-\lim_dP\int_{\tau^c}\Pd(F)dX\\
&=\lim_dP\int_{\tau^c}\Pd(F)dA&&\text{by \eqref{DM}}\\
&=\lim_dP\int_{\tau^c}\Pd(\condP{F}{\F_\tau})dA\\
&=P(A_\tau\condP{F}{\F_\tau})
\end{align*}
proving \eqref{natural}and that there is a version of $A_\tau$ which is $\F_\tau$ measurable
and thus $A$ to be a stochastic process. $A$ is of class $DM_s$ since $M$ is so: if $d\in\D$ then
$S^d(M)=0$. Conversely, if \eqref{DM} holds and $A$ is of class $DM_s$ then so necessarily is
$X$, as was just shown. If $\condP{N}{\F_\tau}-B_\tau$ were another decomposition such as
\eqref{DM}, then $A-B$ would be a uniformly integrable martingale meeting \eqref{natural} and
thus such that $P((A_\tau-B_\tau)\set{F})=0$ for all $F\in\F$ and $\tau\in\T^p$. Eventually, if $X$
is a supermartingale $F\in\F$ and $\tau\leq\upsilon$ then
$\alpha^d_{X,\upsilon}(F)-\alpha^d_{X,\tau}(F)=\phi_X^p(\set{\tu}\Pred^d_P(F))\geq0$ as
$\phi^p_X$ is positive.
\end{proof}

Differently from Meyer \cite{meyer} and similarly to the original result of Doob, decomposition
\eqref{DM} does not require the supermartingale property. Property \eqref{natural} of the
intervening compensator $A$ may be regarded as a version of the classical definition of a
\textit{natural} process. $A$ may however not be predictable as the relationship between these
two properties is unclear outside of the classical setting.

The main drawback of the strong Doob Meyer property defined above, however, is its measure
theoretic nature and, consequently, the  intrinsic difficulty to translate it into a corresponding
property on the process $X$. Thus, establishing sufficient conditions for the existence of the limit
$\mu_X$ seems to be an open problem. Observe, however, that the collection of subsets of $\D$
of the form $\{d'\in\D:d'\geq d\}$  for $d\in\D$ is a filter base. Fix some ultrafilter $\U$ containing
it \cite[lemma I.7.11, p. 30]{bible} and assume that the set $\mathcal S(X)$ is bounded in $L^1$. Then
for each $F\in\F$, the collection of sets of the form $\{\mu_X^d(F):d\in U\}$ for all $U\in\U$ is
itself an ultrafilter in an interval and thus converges, \cite[lemma I.7.12, p. 30]{bible}: write
$\mu^\U_X(F)=\U-\lim\mu^d_X(F)$. This way of writing simply amounts to saying that for each
$\varepsilon$, the set $\{d\in\D:\abs{\mu_X^d(F)-\mu^\U_X(F)}<\varepsilon\}$ is in $\U$.

The following property is the closest relative, in our framework, to the property $D$ considered
by Meyer \cite{meyer}. It shares with that original definition the honest judgment expressed by
Meyer himself who considered it \quot{(...) not easy to handle} (p. 193) and who admitted that
\quot{(...) no satisfactory condition is known, implying that sufficiently general classes of right
continuous supermartingales are contained in the class (D)} (p. 195).

\begin{definition}
\label{def DM}
A stochastic process $X$ is said to possess the Doob Meyer property, or to be of class $DM$, if
$\mathcal S(X)$ is a relatively weakly compact subset of $L^1$.
\end{definition}

The preceding definition may be cast in terms of stopping times. If $\tau_n\in\T^p$,
$F_n\in\F_{\tau_n}$ for $n=1,\ldots,N$ and $F_n\cap F_m=\emp$ then it is natural to define
the simple stopping time $\sigma$ and the value $X_\sigma$ of $X$ at $\sigma$ as
\begin{equation}
\label{X sigma}
\sigma=\sum_{n=1}^N\tau_n\set{F_n}  \quad\text{and}\quad
X_\sigma=\sum_{n=1}^NX_{\tau_n}\set{F_n}+X_\emp\set{\bigcap_nF_n^c}
\end{equation}
Let $\Sigma$ be the collection of all simple stopping times and
$D(X)=\{X_\sigma:\sigma\in\Sigma\}$. The following Lemma provides evidence of the strict
relationship between Definition \ref{def DM} and the class $D$ property (at least as long as
supermartingales are concerned). We consider a slightly weaker version of this property,
defined as follows

\begin{definition}
A stochastic process $X$ is said to be of class $D$ if $D(X)$ is a relatively weakly compact subset of
$L^1$.
\end{definition}
We start noting that if $\{Y_n:n=0,\ldots,N\}\subset L^1_+$, $Y_0=0$ and
$Z_n=\sum_{k=1}^nY_k$, then
\begin{equation}
\label{pre meyer}
\begin{split}
Z_N\sset{Z_N>2k}&=(Z_N-2k)^++2k\sset{Z_N>2k}\\
&\leq(Z_N-2k)^++2(Z_N-k)^+\\
&\leq3(Z_N-k)^+\\
&=3\sum_{n=1}^N\left[(Z_n-k)^+-(Z_{n-1}-k)^+\right]\\
&\leq3\sum_{n=1}^NY_n\sset{Z_n>k}
\end{split}
\end{equation}

\begin{lemma}
\label{lemma meyer}
Let $\T^p$ be linearly ordered. A submartingale is of class $DM$ if and only if it
is of class $D$.
\end{lemma}

\begin{proof}
Of course, if $X$ is a submartingale, then $S^d(X)\geq0$. Assume that $X$ is of class $D$. By the
linearity of the order on $\T^p$, we may assume that each $d\in\D$ is of the form
$d=\{\tau_n\cap\tau_{n+1}^c:n=1,\ldots,N\}$ with $\tau_n<\tau_{n+1}$. Let
$F_n=\{\sum_{j=1}^n\condP{X_{\tau_{j+1}}}{\F_{\tau_j}}-X_{\tau_j}> k\ge
\sum_{j=1}^{n-1}\condP{X_{\tau_{j+1}}}{\F_{\tau_j}}-X_{\tau_j}\}$ and $\sigma_k=
\sum_{n=1}^N\tau_n\set{F_n}$. \eqref{pre meyer} implies
\begin{equation}
\label{D to DM}
P\left(S^d(X)\sset{S^d(X)>2k}\right)\leq3P\left((X_{\tau_{N+1}}-X_{\sigma_k})\sset{S^d(X)>k}\right)
\leq6\sup_{\sigma\in\Sigma} P\left(\abs{X_\sigma}\sset{S^d(X)>k}\right)
\end{equation}
Choose $k=0$ in \eqref{D to DM}: $\mathcal S(X)$ is norm bounded in $L^1$ and therefore
$\lim_k\sup_{d\in\D}P(S^d(X)>k)=0$. Since $D(X)$ is uniformly integrable, then so is $\mathcal S(X)$.
Conversely, assume that $X$ is of class $DM$ and thus, being a supermartingale, a quasi
martingale. Let $\sigma=\sum_{n=1}^N\tau_n\set{F_n}$ with
$\tau_1<\tau_2<\ldots<\tau_{N-1}<\emp$ and $F_N=\bigcap_{n=1}^{N-1}F_n^c$. Set
$\bar F_0=\emp$, $\bar F_n= \bigcup_{j=1}^nF_j$ and
$d=\{\tau_n\cap\tau_{n+1}^c:n=1,\ldots,N-1\}$. Choose $\tau_N\in\T^p_{\tau_{N-1}}$ arbitrarily.
Then, since $\bar F_n\cap\{X_\sigma>k\}\in\F_{\tau_n}$ we have
$$
X_\sigma\sset{X_\sigma>k}=X_\emp\set{F_N}\sset{X_\sigma>k}
+X_{\tau_N}\set{\bar F_{N-1}}\sset{X_\sigma>k}
-\sum_{n=1}^{N-1}(X_{\tau_{n+1}}-X_{\tau_n})\sset{X_\sigma>k}\set{\bar F_n}
$$
and likewise
$$
-X_\sigma\sset{X_\sigma<-k}=-X_\emp\set{F_N}\sset{X_\sigma<-k}
-X_{\tau_N}\set{\bar F_{N-1}}\sset{X_\sigma<-k}
+\sum_{n=1}^{N-1}(X_{\tau_{n+1}}-X_{\tau_n})\sset{X_\sigma<-k}\set{\bar F_n}
$$
In other words, $P(\dabs{X_\sigma}\sset{\dabs{X_\sigma}>k})\le
P\left(\sset{\dabs{X_\sigma}>k}(\dabs{X_\emp}+S^d(X)+\dabs{X_{\tau_N}})\right)$ where
$d=\{\tau_n\cap_{n+1}^c:n=1,\ldots,N-1\}\in\D$.
Since $\tau_N$ may be chosen arbitrarily large, we conclude from Corollary \ref{corollary riesz}
\begin{equation}
\label{DM to D}
P(\dabs{X_\sigma}\sset{\dabs{X_\sigma}>k})\leq P\left(\sset{\dabs{X_\sigma}>k}
(\dabs{X_\emp}+\dabs{M})\right)+\sup_{d\in\D} P\left(\sset{\dabs{X_\sigma}>k}S^d(X)\right)
\end{equation}
Since $X_\emp,M\in L^1$ and $\mathcal S(X)$ is bounded in $L^1$, we deduce that $D(X)$ is also
bounded in $L^1$ and uniformly integrable too.
\end{proof}

Not only does the above property provide a natural extension of the class $D$ property, at least
as far as submartingales or supermartingales are concerned (the claim may in fact fail outside
of this class); it is also the more convenient one in order to prove the existence of Doob Meyer
decompositions. In fact,

\begin{theorem}
\label{th DM U}
A stochastic process $X$ is of class $DM$ if and only if for each ultrafilter $\U$ on $\D$ there is
a decomposition
\begin{equation}
\label{DM U}
X_\tau=\condP{M^\U}{\F_\tau}-A^\U_\tau\qquad \tau\in\T^p,\ P\ a.s.
\end{equation}
where $M^\U\in L^1$ and $A^\U$ is a process of class $DM$ such that $A^\U_\emp=M^\U$ and
\begin{equation}
\label{natural U}
P(A^\U_\tau\set{F})=\U-\lim P\int_{\tau^c}\Pd(F)dA^\U\qquad\tau\in\T^p, \ F\in\F
\end{equation}
The decomposition \eqref{DM U} is unique. If $X$ is a supermartingale and
$\tau\leq\upsilon\neq\emp$ then $A^\U_\tau\leq A^\U_\upsilon$ $P$ a.s..
\end{theorem}

\begin{proof}
The proof is essentially the same as that of Theorem \ref{th DMs}. In fact we get from
\eqref{mu-alpha} the decomposition
$P(X_\tau\set{F})=\mu_X^\U(\condP{F}{\F_\tau}F)-\alpha_{X,\tau}^\U(F)$ for all
$F\in\F$ and $\tau\in\T^p$, with $\alpha^\U_{X,\tau}=\U-\lim\alpha^d_{X,\tau}$. Looking at
the collection $\{\mu^d_X(F):d\in\D\}$ as a function mapping $\D$ into $\R$, it is easy to
conclude that ultrafilter limits are additive and monotone. This proves that indeed $\mu^\U_X$
is an additive set function. Choosing $k$ large enough so that
$\sup_{d\in\D}P(\abs{S^d(X)}\set{\{\abs{S^d(X)}>k\}})<\varepsilon$, we
conclude by monotonicity that if $h\in\Sim(\F)$ with $\abs{h}\leq1$, then
$\abs{\mu^\U_X(h)}\le\varepsilon+kP(\abs{h})$. In other words $\mu^\U_X\ll P$.
Then, $\alpha^\U_{X,\tau}\ll P$ by \eqref{mu-alpha}. Decomposition
\eqref{DM U}, property \eqref{natural U}, adaptness of $A$ and uniqueness then easily follow.
Conversely, if \eqref{DM U}  holds with $A$ being of class $DM$, then the same must be true of
$X$ since $S^d(X)=-S^d(A^\U)$ for all $d\in\D$.
\end{proof}

The main insight provided by Theorem \ref{th DM U} seems to be that, in the context considered
here, there exists a multiplicity of suitable compensating schemes, each involving a different notion
of naturality the choice among which appears as rather arbitrary.


TITLE: What happens when a force is applied to a flux pinned superconductor?
QUESTION [3 upvotes]: When a force is applied to a superconductor to move it in a direction that it resists because it is flux pinned, where does the additional energy used to overcome the flux pinning resistance go? E.g. is it dissipated as radiation, current, lateral/rotational motion or something else?

REPLY [3 votes]: The energy you supply by the force is transformed into changing the flux tube configuration. Microscopically this means that certain parts of the superconductor go from superconducting to "normal" in a small areas (flux tubes) to allow the magnetic flux through and the energy supplied is used to rearrange these regions. In superconductors that stay fixed above a magnet the flux tubes are generally stuck to defects in the lattice. In this case it costs energy to move the superconductor away since energy is needed to create flux tubes not near a defect.
As a side note, I love STM movies of flux tubes: https://www.youtube.com/watch?v=vwFm7d_0GsA


TITLE: Integral of logarithm of cotangent divided by squared sine.
QUESTION [0 upvotes]: My task is:
$\int\frac{\log(|\cot x|)}{\sin^2x}\,dx$
If i try using the formula:
$\int f(x)*g'(x) dx = f(x)*g(x) - \int f'(x)*g(x)dx$
At some point it throws out
$\int \tan(x)\,dx$
which even Wolfram says is incalculable on my basic university level.
I can't come up with any good substitution method. Any ideas? Thanks.

REPLY [0 votes]: Are you sure Wolfram says $\displaystyle\tan x\,\mathrm d x$ is incalculable? It is simply, by the substitution $u=\cos x$, $\color{red}{-\ln(\cos x)}$.
Here, again the substitution
$$u=\cot x,\quad\mathrm d u=-\frac1{\sin^2x}\,\mathrm dx,$$
(dropping the absolute value, since the logarithm requires the argument be positive) leads to to a well-known integral
$$\int\frac{\ln(\cot x)}{\sin^2x}\,\mathrm dx =-\int\ln u\,\mathrm du= -(u\ln u-u).$$


TITLE: Relevancy to GCD and LCM
QUESTION [1 upvotes]: The traffic lights at three different road crossings change after every 48 sec, 72 sec, and 108 sec respectively. If they all change simultaneously at 8:20:00hours, then at what time will they change again simultaneously?
okay seriously how would someone even guess we have to do this by LCM GCD method ?

REPLY [0 votes]: "okay seriously how would someone even guess we have to do this by LCM GCD method ?"
If the first light blinks ever $48$ seconds then any time it blinks will be at a mmultiple of $48$.  So the answer when the all blink will have to be a multiple of $48$ because those are the only times the first one blinks.
The second only blinks at a multiple of $72$ seconds.  So the answer has to be a common multiple of both $48$ and $72$.
And so on.  The answer has to be a common multiple of all $48$,  $72$, and $108$.  What is a common multiple of $48$, $72$, and $108$.
Well... notice the first light will blink at: $48, 96, 144, 192, 240, 288, 336, 384, 432, 480,......$
The second light at: $72, 144, 216, 288, 360, 432, 504, 576, 648, 720 ....$
The third at: $108, 216, 324, 432, 540, 648, 756, 864, 972, 1080 ....$
What number do they all have in common?  Well.... I can see it is $432$.
Dang.  That was tedious.  Maybe there is an easier way to figure out what is the least common mulitple of $48, 72$ and $108$.


TITLE: Determine if function is convex
QUESTION [2 upvotes]: I want to determine if the function
$$f(x,y)=e^x+e^y+x^2-2xy+4y^2+2x+3y-8$$
is convex on $\mathbb{R}^2$. Any ideas on how to prove convexity for two variable functions?

REPLY [0 votes]: $$f(x,y)=e^x+e^y+x^2-2xy+4y^2+2x+3y-8$$
$$\nabla f=\begin{bmatrix}  e^x+2x-2y+2 \\ e^y-2x+8y+3\end{bmatrix} $$
$$\nabla^2 f=\begin{bmatrix}  e^x+2 & -2 \\ -2 & e^y+8\end{bmatrix} $$
Since
$$e^x+2>0$$
and $$(e^x+2)(e^y+8)-2^2=8e^x+2e^y+16-4=8e^x+2e^y+12>0$$
The function is convex.
Another way to see this is to observe that since
$$\nabla^2 f=\begin{bmatrix}  e^x+2 & -2 \\ -2 & e^y+8\end{bmatrix} =\begin{bmatrix}  e^x & 0 \\ 0 & e^y\end{bmatrix}+ \begin{bmatrix}  2 & -2 \\ -2 & 8\end{bmatrix}.$$
Observe that
$\begin{bmatrix}  2 & -2 \\ -2 & 8\end{bmatrix}$ is positive semidefinite from Gershgorin circle theorem, the eigenvalues lie between $0$ and $10$.
Also, $\begin{bmatrix}  e^x & 0 \\ 0 & e^y\end{bmatrix}$ is positive definite.
Hence $\nabla^2f$ is positive definite.