<human>: Give me the gist of the text in a nutshell

My girlfriend has a close relationship with her ex.

They were together for a 3 years. She broke up with him. He knows about me. We've been together for coming up to 5 months.

I know they text regularly. And I am fine with that. However, to me it appears as though he is still not over her and is playing the chasing/ i'm going to always be there for you in the hopes of you coming back to me - game. I say this because he recently texted her a "morning beautiful!" and continued their conversation. I was next to her when it happened but she didn't see. I don't think that type of interaction is appropriate at all. 

It's been a recurring theme - her past/baggage in the relationship and how I react to it. She's been open to me about it from the start. But it's small things like this that get to me. He has other examples of not being over the break up.

We had an emotional discussion about this all. I didn't tell her I saw your messages as I don't want to be snooping on her conversations. But I later expressed my view that I hope he respects our relationship and she said he does. It was a difficult discussion to have and I don't think we are any better off and made things worse.

Where do I go from here? I love the girl and want it to work. She see's me getting upset and not over things and finds it distressing that I am like that. I'm coming off as insecure and am of the belief of once the relationship is over you move on from each other. So it's been difficult me having to deal with this. I've had to compromise on this issue with her.
<bot>: girlfriend maintains close relationship with her ex. I am of the complete opposite view but am having to compromise on this issue. We argue over things  like that.