Is the codomain of random variable $X$ always equal to the domain of the associated p.d.f. (or p.m.f.)?

Question

A random variable $X$ is a function from the sample space $\Omega$ to a measurable space $E$.

Is the codomain of random variable $X$ always equal to the domain of the associated p.d.f. $f$ (or p.m.f.)?

I think this must be the case in order for the notation $f(X)$, which is supposed to mean the composition of $f$ and $X$, to make sense.

This question arises after having gotten an answer to the other question: Can we really compose random variables and probability density functions?.

Answer 1

A random variable need not have a p.d.f.. If it has, then the p.d.f $f$ is by definition a measurable function defined on the entire real line such that $P(X\leq x)=\int_{-\infty}^{x} f(t)dt$ for all $x$. Therefore $f(X)$ makes sense irrespective of the actual values attained by $X$. However $f$ is unique only up to sets of measure $0$ so for $f(X)$ to make sense we have to fix one measurable version of $f$.