I am looking for references for the following problem: I have a very special class of regular languages and my aim is to express (and to justify my conjecture) that this class itself is very small in some way (as a subset of the regular languages) and that the languages contained in this class are rather "bloated".
For the latter point, I could prove that all languages in the class have a large diameter with respect to many common metrics on strings. However, I want to consider the following: Given a language from the class, we know it has a large diameter, but does it also have a large "volume" (that is, measure), or put differently, if I choose randomly a finite word, is there anything meaningful to say about how "probable" it is that the word belongs to the language? Of course, we can lift the problem: Picking a random language, how probable is it to get a language in the class?
Are there any references or standard approaches for looking at classes of (regular) languages from this point of view (or is this considered as generally uninteresting)?