I'm scared of coming out to my boyfriend - long post but please read! I need advice!

At the start of our relationship my boyfriend expressed his views that bisexual people just need to 'pick' a sexuality. He got these views from his mother screwing into his brain that bisexual people are greedy for wanting to date both genders...

I've had a lot of deep conversations with him since then and he now says that he doesn't think this way anymore and that he wouldn't care if I came out as bisexual (I asked him what he would think if I was bisexual), and even made jokes that he'd love it because he would find it got seeing me 'do things' with another girl (which I found slightly offensive because being bisexual doesn't mean I'm not monogamous)...

Anyway, as you've gathered I am bisexual and nobody knows. I'm afraid to come out to him because what if somewhere in his brain or subconscious he still does have these harmful thoughts about bisexual people.. I'm also scared of what his mother would think.

I'm not scratching to come out or anything. But I do feel bad for him not knowing this about me. Does anyone have advice on what I should do, or if I should just not tell him?