Why $P(X=Y)=0$ if $X$ and $Y$ are i.i.d continous r.v.s? I read this conclusion in text book:
If $X$ and $Y$ are i.i.d continuous r.v.s, then $P(X=Y)=0$, but it doesn't give any proof. I am a bit confused here, what is the rational for this conclusion? Is it because for a continuous r.v., the probability of it equaling to any particular value is $0$?
Thanks.

For a legalistic argument: $P(X=Y)=E( \mathbb 1_{X=Y}) = E( E( \mathbb 1_{X=Y}|Y)) = E( 0) = 0$ since $E(\mathbb 1_{X=Y}|Y) = 0 \text{ a.s.}$.  That is, the OP's observation that continuous random variables take on particular values with probability zero, when used with Fubini's theorem (a.k.a. the law of total expectation), does the trick.
