Well, I have some experience with the same problem. My ex, whom I adored very much, was a compulsive liar and an eventual cheater. I know how painful this experience is for you. When I found out he was emailing with another woman, I didn't tell him I read his emails, but eventually it was obvious he was infatuated with someone else. When I confronted him, he lied and denied it, then changed his email password. During this whole time he acted like a loving, caring husband, but eventually he really did cheat for real, and from then it spiraled into multiple affairs. It was as if he felt more confident after the first time he got away with cheating, that he just kept on going for more.
I was so in love with him, that I wanted nothing more then to preserve the marriage. I put up with a lot of pain, but also I lost a lot of respect for myself. Eventually, he divorced me stating that he didn't love me any more. This was after 12 years of marriage and the height of my love for him. I hate to tell you this, and I really hate to make you feel worse than you already feel, but if a man wants to cheat, he will, and there's nothing you can do about it. Sounds to me like you married a dishonest person to whom self-gratification is more important than the sanctity of marriage.
I can't tell you what to do, but I want to give you some advice. The biggest lesson I learned from this experience is to be true to myself. I wish I hadn't compromised my goals, my career, my ethics and my self-respect for his benefit. I lost myself in that marriage, and when I was finally left alone, I no longer had any idea who I am as a person. It has taken me years to get back in touch with myself. Sometimes when we fall in love, we do so blindly. We believe in love so much, that we look the other way when something isn't right.
Before you convince yourself that he will change (no, he'll only learn to lie better), take some time and think about what you want. What is best for you and the children. In the end, the choice may not even be yours.
Sorry, I don't mean to throw oil into your fire, I just wanted to let you know you have every right to be angry, and to expect better.
xo