Forgiving him depends on a lot of factors. If he is truly remorseful and willing to work on the relationship, it might be possible. But it won't be easy and will take time.
If you've caught your husband in such a situation, first, take some time to calm down. It's a very shocking discovery. You could consider having an open and honest conversation with him about why he did it, your feelings, and what it means for your relationship.
One common story is that they might be a bit nervous at first. They would share some soft words, maybe talk about how they are feeling excited and a bit scared at the same time. Then they would slowly start to get to know each other more intimately in a very gentle and loving way.
There is no evidence to suggest that such activities occur in a gym environment. Gyms have rules and ethics in place to maintain a proper and respectful atmosphere for all members. This kind of description violates the basic norms of a public fitness place.