Let $(X,\mathcal{F},\mu)$ be a measure space. Let $(Y,\mathcal{G})$ be a measurable space. Let $T:(X,\mathcal{F})\to(Y,\mathcal{G})$ measurable. Let $f:(Y,\mathcal{G})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ measurable. We have $$\int\limits _{Y}f \, dT_{\#}\mu=\int\limits _{X}(f\circ T) \, d\mu$$

Let $X$ be a rv on $(\Omega,\mathcal{F},\mathbb{P})$ such that $X\sim\mu$. Then for any Borel function $f:\mathbb{R}\to \mathbb{R}$ we have $$\mathbb{E}[f(X)]=\int\limits _{\Omega}f(X(\omega)) \, d\mathbb{P}(\omega) =\int\limits _{\mathbb{R}}f(x) \, d\mu(x)$$provided either side is well-defined.