I know the question is a little vague; I guess it sort of has to be.
So, we know that there are subsets of $\mathbb{R}$ that fail to be measurable. I was wondering if there's a way to get a sense of "how many" such sets their are.
I'm thinking that we'd need to define a measure on some sigma algebra in $\mathcal{P}(\mathcal{P}(\mathbb{R})),$ so that our ambient space is the powerset of the reals, and the elements of the sigma algebra will then be collections of subsets of real numbers. From there, define some suitable notion of measure and get a sense of "how many" subsets of real numbers are (say lebesgue) measurable.
Does anyone know if this has been done, or if it's even possible? For all I know, there might be some deep problems with trying to construct such a measure.
Maybe a slightly different way of phrasing my question, though similar in spirit, is; If I were to 'randomly' select a subset of $[0,1]$, what's the probability that it would be measurable, if such a probability exists?
