$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ > [!theorem] Change of Variables > Suppose we have maps $(\Omega, \FFF, \PP) \stackrel{X}{\to} (S, \GGG, \mu) \stackrel{f}{\to} (\RR, \BBB, \nu)$. Let $Y = f\circ X$. In other words, $X$ is a random variable, and $Y = f(X) \in \RR$ is a function of $X$. Suppose $f\geq 0$ or $Y$ is integrable. Then, > $ \EE[Y] = \int_S f\der \mu $ > i.e. > $ \int_\Omega f(X(\omega)) \der \PP(\omega) = \int_S f(x) \der( X_\sharp \PP)(x).$ In particular, one can take $(S, \GGG) = (\RR, \BBB)$ and $f(x) = x$; then $ \EE[X] = \int_\RR x \der (X_\sharp \PP)(x)$ as expected. # Proof This is our first major usage of the [[Convergence properties of Lebesgue Integral|convergence theorems]]; we will follow the [[measure theory function scaffold|standard scaffold]]. > [!part]- Indicators > If $f = \id_E$ with $E$ measurable, then both sides each $\PP(X\in E)$. > [!part]- Simple Functions > If $f = \sum_{i=1}^n c_i \id_{E_i}$, we apply linearity to show both sides equal $\sum_{i=1}^n c_i \PP(X\in E_i)$. > [!part]- Nonnegative Functions > [[real-valued measurable functions#^sf-approx|One can find]] simple functions $f_n\uparrow f$. Then, by [[Convergence properties of Lebesgue Integral#^Monotone-Convergence|Monotone convergence]], we conclude that both sides are the limit > $\lim_{n\to \infty} \int_\Omega f_n(X(\omega)) \der \PP.$ >[!part]- Absolutely Integrable >Similarly, [[real-valued measurable functions#^sf-approx|one can find]] simple functions $f_n\to f$ such that $\abs{f_n} < \abs{f}$. Then, by [[Convergence properties of Lebesgue Integral#^Dominated-Convergence|Dominated Convergence]], we conclude that both sides are the limit > $\lim_{n\to \infty} \int_\Omega f_n(X(\omega)) \der \PP.$