Measures and probability in formal language theory
There is the concept of density of languages (see e.g. here). The density $\operatorname{den}_L : \mathbb{N} \to [0,1]$ of $L \subseteq \Sigma^*$ is defined by

$\qquad \operatorname{den}_L(n) = \frac{|L \cap \Sigma^n|}{|\Sigma^n|}$.

For any fixed length, the density corresponds to the probability of picking a word from the language, assuming we pick uniformly at random. Add a distribution over lengths and you may have what you need.

You may be able to express your concept of "volume" in terms of this notion, maybe by investigating $\lim_{n \to \infty} \operatorname{den}_L(n)$.

As for "randomly picking a language" -- how would you do that? There are uncountably many languages over any given alphabet so I'm not sure how you would define a (nice) probability distribution.