TITLE: Probability of selecting a non-measurable set
QUESTION [12 upvotes]: If you randomly select a subset of $[0,1]$, what is the probability that it will be measurable?
Edit: This question may be unanswerable as asked.  If additional assumptions could be made to make it answerable, I would be interested in learning about them.

REPLY [2 votes]: I am assuming that we have fixed some $\sigma$-field of subsets of the interval and "measureable" subset means a subset from that $\sigma$-field.
Your question is meaningless until we specify what does "randomly select a subset" mean. And to do that we have to choose a $\sigma$-field of subsets of the set $2^{[0,1]}$, say $\mathcal{A}$, and a probability measure $\mathbb{P}:\mathcal{A}\rightarrow [0,1]$. It is easy to see that now when we know what does "randomly select a subset" mean we can answer your question. And what is the answer? Depending on the choice of $\mathcal{A}$ and $\mathbb{P}$ it can be any number between $0$ and $1$, and none of those answers is better than the other. (From the point of view of probability theory)