 Probability density of a product of random variables

If $(X,Y)$ has a joint probability density $f$, the product has the displayed probability density function, interpreted almost everywhere. For fixed $x\ne 0$, substitute $z=xy$ in the integral defining $\mathbb{P}(XY\in A)$; the absolute Jacobian determinant is $|x|$. The Tonelli theorem then exchanges the nonnegative integrals. The line $x=0$ has two-dimensional Lebesgue measure zero, so its omission creates no probability mass. A density may be infinite at exceptional points without creating an atom.