I am having lots of trouble proving the following statement:
Let $X,Y$ be two real valued random variables on a probability space $(\Omega,\mathcal{F},P)$. These two variables are independent and identically distributed, iid, in the sense that they share the same distribution $P_X=P_Y$ and the cumulative distribution function $F_X(x)=P_X[X\leq x]$ is continuous. The probability measure $P$ might not necessarily be absolutely continuous wrt a sigma-finite measure. Therefore there might not be a Radon-Nikodym derivative with respect to a sigma-finite measure. If there was a Radon-Nikodym derivative with respect to a sigma-finite measure, it is easier to show what I want.
I need to prove that, under the above conditions $P[X=Y]=0$
Any insights?
Best regards,
Juan Manuel