Does $E(XY)=E(X)E(Y)$ always hold when $X$ and $Y$ are independent?

Question

Let $X$ and $Y$ be independent random variables. The standard proof for $$E(XY)=E(X)E(Y)\tag{$\ast$}\label{exy}$$ either uses $$E(X)=\sum_xxf(x)$$ if $f(x)$ is a pmf of $X$, or $$E(X)=\int_{-\infty}^\infty xf(x)\,dx$$ if $f(x)$ is a pdf of $X$. But what if $X$ or $Y$ is a continuous random variable without pdf? Does \eqref{exy} still hold, provided that $E(XY)$, $E(X)$, and $E(Y)$ all exist?