On transformation of r.v.s. and sigma-finite measures

[psie]

TL;DR Summary

I'm reading an article on transformation of random variables. In the article they restrict to $\sigma$-finite measures, but I don't understand why.

I'm reading this article on transformation of random variables, i.e. functions of random variables. We have a probability space $(\Omega, \mathcal F, P)$ and measurable spaces $(S, \mathcal S)$ and $(T, \mathcal T)$. We have a r.v. $X:\Omega\to S$ and a measurable map $r:S\to T$. Then we want to find the distribution of $r(X)$ given that of $X$. Pretty soon into the article, after the first proposition, under the very first diagram, they say that we should then consider $\sigma$-finite measures on $S$ and $T$. I don't understand why we need to restrict to $\sigma$-finite measures. What necessitates this?

 

[psie]

Ok, I guess you can ignore the question. I believe it is because of the existence of density functions, if I'm not mistaken.