Change of Variables Formula
Here's the standard change of variables formula (notation and assumptions in the first comment) for a test function $\varphi:\mathsf{Y}\to[0, +\infty]$ and a pushforward measure $\mathbb{P}\circ f^{-1}$ by a measurable function $f$. $$ \int_{\mathsf{Y}} \varphi \, d\mathbb{P}\circ f^{-1} = \int_{f^{-1}(\mathsf{Y})} \varphi\circ f \, d\mathbb{P}. $$
Question
Suppose rather than $P_f$ I have a distribution $\mathbb{P}\circ g$, for a function $g:\mathsf{Y}\to\Omega$. Does the change of variables formula still work? $$ \int_{\mathsf{Y}} \varphi \,d \mathbb{P}\circ g = \int_{g(Y)} \varphi\circ g^{-1} \, d\mathbb{P} $$