Proof of Change of Variables for Expectations

$\varphi\circ X$ is a random variable: for a Borel set $B$, $(\varphi\circ X)^{-1}(B)=X^{-1}(\varphi^{-1}(B))\in\mathcal{F}$, since $\varphi^{-1}(B)$ is Borel by measurability of $\varphi$ and $X$ is measurable.

Step 1 (indicators and simple functions). For $\varphi=\mathbf{1}_B$ with $B$ Borel, $\varphi\circ X=\mathbf{1}_{\{X\in B\}}$ and, by the integral of a simple function on both sides,

By linearity for nonnegative functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), the identity extends to every nonnegative simple $\varphi$ with Borel level sets.

Step 2 (claim 1). Let $\varphi\ge 0$ be Borel measurable. With the dyadic functions $\varphi_m(t)=\min\{m,2^{-m}\lfloor 2^{m}t\rfloor\}$ of Step 0(b) of the proof of Linearity and Monotonicity of the Lebesgue Integral, the compositions $\varphi_m\circ\varphi$ are nonnegative simple functions on $\mathbb{R}$ with Borel level sets (preimages under $\varphi$ of the Borel level sets of $\varphi_m$), nondecreasing to $\varphi$ pointwise; likewise $\varphi_m\circ\varphi\circ X$ increases to $\varphi\circ X$. Applying Monotone Convergence Theorem on $(\Omega,\mathcal{F},P)$ and on $(\mathbb{R},\mathcal{B}(\mathbb{R}),P_X)$ and using Step 1 at each stage,

Step 3 (claim 2). For general Borel $\varphi$, apply Step 2 to $|\varphi|$: $\int_\Omega|\varphi\circ X|\,dP=\int_{\mathbb{R}}|\varphi|\,dP_X$, so by Integrable Function and the Lebesgue Integral the function $\varphi\circ X$ is integrable with respect to $P$ exactly when $\varphi$ is integrable with respect to $P_X$. In that case, applying Step 2 to the positive and negative parts $\varphi^{\pm}$ (Borel measurable by Integrable Function and the Lebesgue Integral, with $\varphi^{\pm}\circ X=(\varphi\circ X)^{\pm}$ pointwise) and subtracting gives the identity in $\mathbb{R}$.

Final remark. Expectations, moments, and variances are integrals of Borel functions of the variable ($t\mapsto t$, $t\mapsto t^{k}$, $t\mapsto(t-\mathbb{E}[X])^{2}$, each continuous, hence Borel by the generator criterion of Measurable Function and Real-Valued Measurable Function), so by claims 1 and 2 they are determined by the distribution; identically distributed random variables therefore share them. $\blacksquare$