closeness of a set of probability distributions how can one check if a family of distributions is closed? what does closeness of a set of distributions mean? I understand the concept of closed sets in the context ofEuclidean geometry, but I have difficulty understanding what these concepts mean for a set of probabilities.

It all depends (by definition, really) on what topology you put on the set of distributions. There are probably several that are used (weak topology, vague topology IIRC), depending on context. So go find out what that context is and you'll know what closed means. It probably won't be easy, depending on how strong your background in general topology or functional analysis or measure theory is.
