I've been thinking for a while about the following question, and I cannot make up my mind. Any ideas are welcome about it:
I have a set function $V:\Omega\to 2^{[0,1]}$, and ideally, I would want to define $V$ as a random variable. My crux is that for sure I know I cannot define the probability measure over $2^{2^{[0,1]}}$, just because of the same reason we cannot do that on $2^\mathcal{R}$. By analogy with the real valued case, I've been thinking on using $\mathcal{B}(2^{[0,1]})$ as my sigma algebra to define the probability measure, however, I'm not sure it is "small" enough to work, with Absolutely Continuous distributions.
Any thoughts about it?