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?

Thanks in advance.


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