Why $P(X=Y) = 0$ when $X$ and $Y$ are identically distributed, independent, continuous random variables?

Question

If $X$ and $Y$ are identically distributed, independent, continuous random variables, then $P(X=Y) = 0$.

I know that for any particular value $x$, $P(X=x)=0$, but how to show the above rigorously?

Answer 1

Hint: $$P(X = Y) = E[I\{X = Y\}] = \int \int I\{x = y\}\, dP^X(x)\, dP^Y(y)$$

Answer 2

Let $\epsilon\to0$, $F,f$ be the CDF and PDF of r.v. $X$. From the total probability theorem: $$ P(X-Y\le \epsilon)=\int_{-\infty}^\infty P(X-t\le\epsilon)P(Y=t)dt=\int_{-\infty}^\infty F(t+\epsilon)f(t)dt $$

$$ P(X-Y\le -\epsilon)=\int_{-\infty}^\infty P(X-t\le-\epsilon)P(Y=t)dt=\int_{-\infty}^\infty F(t-\epsilon)f(t)dt $$

$$ P(X-Y\le \epsilon)-P(X-Y\le -\epsilon)\approx2\epsilon\int_{-\infty}^\infty f(t)f(t)dt\to 0 $$

Therefore, the CDF of $X-Y$ is continuous at $0$.