[cdavid]

That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0.

[edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.

[philipov]

oh, yeah, but that's because although Q is dense, it is not a dense subset of R and locally that's equivalent to saying a single point is not dense in R

[cdavid]

no, it is not. The OP statement is about the probability of the event "{X in Q}" (equal to 0), with X a random variable uniformly distributed on a bounded interval. That even contains many points (infinitely many actually), but has a probability 0.

You are talking about the probability of a single point event, which is also always 0 on that same sigma algebra.

The OP point is not completely trivial because the event contains an infinite (but countably) number of elements. It is fairly easy to understand though since by its very definition, the P[{X in Q}] = sum P[{x}] taken over every rational number (since Q is countable), and each P[{x}] is 0.

A deeper statement is that there exists uncountable sets of probability 0.