probability - Basic question related to cdf and pdf of random variables

If there are two independent random variables $X$ and $Y$ (there can have only non-negative values) with corresponding pdfs and cdfs as $f_X(x), f_Y(y)$, $F_X(x),F_Y(y)$ respectively. Then in what case the following formula is correct $$\int_0^{\infty}f_Y(y)F_X(y)dy=0$$ This question is raised from a previous question Compute the CDF of $\left[\log_2(1+X)-\log_2(1+Y)\right]^+$ . Thanks in advance.

Answer

You're asking when is $$E[F_X(Y) ] = 0$$

Since $F_X$ is a nonnegative function, we need that $F_X(Y) = 0$ almost surely, that is, that $Y 0\}$$

Note that the independence assumption is irrelevant since $X$ appears only through its CDF $F_X$.