Many results in probability theory/random matrix theory/etc require probability distributions with finite fourth moments; what is the measure of such probability distributions (in the space of probability measures)? While the question "what RVs has finite fourth moment" is answered here, I'm trying to determine how often you would expect "real-life data" to conform to hypotheses in the various results above.

Edit: since this isn't my primary field, my question is naive, apologies. I was primarily interested in the measure of finite moment distributions relative to infinite ones, particularly fourth and lower moments (although I suspect it won't matter). However, thanks to Nate, below, I now know there is no canonical measure on distributions. Since there are no canonical measures, my question becomes:

What are some commonly-used measures for the space of distributions, and what proportion do finite moment distributions occupy in them?

Additionally, because there aren't canonical measures, I'd like to know if these finite moment distributions are dense in two particular topologies, the ones induced by $L_p$ and Fisher metrics (generally divergence-induced metrics). (I haven't though about how the two metrics are related topologically, if at all).