Change of variables Expectation

Question

For a measurable map $X:\Omega\to S$ and a real map $g:S\to R$, I am trying to rewrite the integral $\int_A g(X(\omega))dP(\omega)$ as an integral over $S$. (of course with the usual non-negativity or finite expectation assumption). Furthermore with this new integral, is it possible to express the new measure (over S) in terms of $P, A$ and $X$ only?

Answer 1

Let $\mu$ be defined on $S$ by $\mu (E)=P(X^{-1}(E))$. Then $\int_{X^{-1}(E)} g(X)dP=\int_E g d\mu$. If $A$ is not of the form $X^{-1}(E)$ for some measurabke set $E$ in $S$ we cannot write $\int_A g(X)dP$ as an integral w.r.t. $\mu$ in general.