<human>: Can you condense the text into a few key points?

My boyfriend and I have been together for 3.5 years. We've lived together for almost 3 of those with virtually no problems. However, when he let me use his tablet the other day (yes, let me. He literally handed it to me) I unlocked the screen to see a chat log open, which I immediately tried to close out of without looking at anything. I've done it before, no bid deal. But I'm kind of rubbish with hand-eye coordination, and it took me two tries to close the window, but not before I saw that he had been talking with his ex girlfriend. Normally I wouldn't care, since I still have friendships with a few of mine. But then I saw him talking about how she's still his favorite gf because she always made him cookies, and just general flirting back and forth. To be honest, I can't remember specific words because I felt pretty ashamed for seeing it in the first place, and I managed to close it out pretty quickly after seeing what I saw. I've been pretty concerned because he told me before we started dating that he had been really devastated when she broke up with him.

My concern is this: should I be worried? Should I just ignore it? Even though I didn't purposefully look through his messages, I still feel like I violated some sort of trust. And I can't help but be worried because he's cheated on other girlfriends in the past, so part of me is always going to wonder if it's something he's capable of doing again.
<bot>: BF with history of cheating on past SO has been flirting with his ex, and I'm not sure how to deal with it.