# Probability distribution function for $Z=XY$

Currently, I am reading Probability Theory and Example from Durret.

Here is the question 2.1.14 from the 5th edition.

Let $$X,Y\geq 0$$ be independent with distribution functions $$F,G$$, find the distribution function for $$XY$$.

Here, the question hasn't assume that $$F,G$$ are absolute continuous. The hint I found online was to apply the natural logarithm to $$XY$$, and then we have \begin{align*} \ln(XY)=\ln(X)+\ln(Y) \end{align*} and use the convolution to find the distribution. My question is how can we deal with the case $$X=0$$ or $$Y=0$$? The domain for a distribution function has to be $$\mathbb{R}$$ rather than $$\overline{\mathbb{R}}$$. Is there any kind of rigorous argument about this?