For a while, it will be natural for you to question & doubt your spouse. There is no shortcut around the TIME it takes for trust to be re-established. So, even if the affair is 100% over, it would be natural for you to still wonder. For a while, you are going to be very sensitive to many behaviors that seem suspicious to you, even if they are actually innocent.

In my opinion, a betrayed spouse has the right to be reassured. Your spouse should willingly give you open access to his/her schedule, phone history, text messages, email accounts, etc. Any attempt to keep parts of their lives private will only work against your ability to trust them again. And for a while, it is reasonable to assume that you may need to investigate your suspicions in order to settle your own fears. This can't go on indefinitely, but it may be helpful in the beginning stages of recovery so that you begin to gain more confidence in your spouse's sincerity (or learn the truth about their insincerity). 

But what about those times when you don't have any evidence but you just don't believe your spouse is telling you the truth? In my opinion, it is better for you to act in belief. What I mean is that you should respond to your spouse as though they are telling you the truth (innocent until proven guilty). Is there some risk in this? Yes. Maybe they really are being dishonest. But without evidence, I believe you are better off assuming they are being truthful rather than assuming they are lying. Let me explain why... 

If we compare the likely outcomes of acting in belief versus acting in disbelief, I think the former encourages healing and avoids conflict more than the latter. Consider the likely possibilities:

If you choose to act in belief...

• ...and your spouse is being truthful, then your response is seen as encouraging, trusting, and supportive and will reinforce the kind of choices you want them to keep making.
• ...but your spouse is lying, then they will be doing what they were going to do anyway and eventually you'll find out, experience the pain of betrayal again (I don't mean to minimize that), and make appropriate choices. In the meantime, though, you will have avoided the "going nuts" routine of constant questioning, demanding, etc. that never works in your favor, anyway.

If you choose to act in disbelief...

• ...and your spouse is being truthful, then they will become increasingly discouraged and may, finally, just give up trying. If your spouse is being honest and making sincere efforts to change, being constantly accused of not changing will quickly deplete any hope they had in marriage recovery.
• ...and your spouse is lying, then all your accusing, arguing, and insisting will do nothing to change their behavior. In fact, it is more likely to make them more defensive and secretive. In this scenario, about all you can expect is ongoing conflict.

Here is the one exception to everything I've just said: If your spouse has always struggled with lying (even apart from the affair issues), then you will need to be more cautious. The best predictor of future behavior is past behavior, but past behavior does not guarantee future behavior. People really can change. If your spouse shows a sincere, ongoing attempts to be honest with you, then you will also have to eventually decide between the two choices listed above. It may just take you a bit longer to reach that choice.