Proving that $P(X=Y) = 0$ for any two continuous random variables

Question

I have the following question to prepare for a lecture at Uni but I've been stuck on this for a long time:

Question:

1. Let $Z$ and $V$ be independent with distribution $U[0,1]$. Show that $P(Z=V) = 0$.

   Hint: Cut $[0,1]$ into intervals $A_1, \ldots , A_n$ of length $\frac{1}{2^n}$ and then used the fact that $$\{Z=V\}\subseteq \bigcup_{i = 1}^{2^n} \{Z\in A_i\} \cap\{V \in A_i\}$$

2. Using the fact that $P(Z=V) = 0$, show that $P(X=Y) = 0$ for $X$ and $Y$ being any continuous and independent random variables.

I have proved the first question, however I'm not sure how to solve the second. I know I maybe can use the fact that I can create any random distributed variable from the uniform distribution, lets say $X = F^{-1}(Z)$, but I do not know how to implement this. Any tips?

Answer 1

By Fubini, \begin{gather} P(X = Y) = E[\mathbf 1_{X=Y}]= \iint_{\mathbb R^2} \mathbf 1_{x=y} F_X(dx) F_Y(dy) = \int_{\mathbb R} E[\mathbf 1_{X=y}] F_Y(dy)\\ = \int_{\mathbb R} P(X=y) F_Y(dy) = 0 \end{gather} (the inner probability is zero in view of continuity of $X$). Note that $Y$ does not need to be continuous.