I'm sorry, but I have to disagree with some of the suggestions that the right thing to do is to allow your H access to everything to show him he can trust you again. It sounds like he's really having trust issues along with an attitude that he "owns" you and that he has a right to control your life. Infidelity is a terrible thing, but it doesn't do either one of you any good to continue down this path (you giving up your boundaries and he acting like he owns your life).
Trust needs to be built slowly again, and he has to choose to trust you again, not by thinking he has a right to control everything about you and having access to your emails. That's not trust, that's ownership of another person.
Eliminating the BF is a first step, but it's only a first step. You will have to earn his trust again, and I don't think that will magically happen by allowing him access to your emails. If he's checking up on you constantly (and not trusting that you're telling him the truth), that is not the beginnings of building an honest and trusting relationship again.
And likewise, he will have to earn your trust again, too. And that's not by trying to dominate you or browbeat you by demanding to know your every actions or your every single thoughts.
IMHO, if you give in to his current behavior, it's like putting a band aid on a severed limb. It's the wrong thing to do.
He needs to behave like he wants you as a loving, caring, faithful partner back in his life, not like he wants someone who he has absolute control over. Yes, you did wrong, but that doesn't give him the right to do the things he's doing. If he really wants you back, he needs to understand that he is wanting you back for the right reasons. And you need to know that you are trying to reconcile for the right reasons as well.
Sorry for the babbering, but I just feel that after reading your sitch, that giving in to his unreasonable (IMO) behaviors is probably not in the best interest of re-creating a relationship that's beneficial to the two of you. JMHO. Best of luck.
Me 37; W 35; 2 dogs M 8 years; before that, dated 9 years Bomb #1: 10/13/2006 (day after my birthday) Bomb #2: 1/15/2007 I am finding strength I never thought I had