Change of variables for expectation

By [L2], the composite $g\circ X$ is measurable. If $s={\sum }_{j=1}^m{c}_j{\chi }_{B_j}$ is a nonnegative simple function on $S$ with the $B_j$ pairwise disjoint, then $s\circ X={\sum }_{j=1}^m{c}_j{\chi }_{X^{-1}(B_j)}$ is a nonnegative simple function on $\mathrm{\Omega }$. Using [L1], [L3], and [L4], $\mathbb{E}[s(X)]={\sum }_{j=1}^m{c}_j\mathbb{P}(X^{-1}(B_j))={\sum }_{j=1}^m{c}_j{\mathbb{P}}_X(B_j)={\int }_{S}sd{\mathbb{P}}_X\mathrm{.}$

Assume now that $g\ge 0$. By [L3], choose nonnegative simple functions $s_n\uparrow g$ on $S$. Then $s_n\circ X\uparrow g\circ X$ by step 1.1, so monotone convergence on both spaces and step 1.1 give $\mathbb{E}[g(X)]={\lim }_{n\to \mathrm{\infty }}\mathbb{E}[s_n(X)]={\lim }_{n\to \mathrm{\infty }}{\int }_S{s}_{n}d{\mathbb{P}}_X={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Assume $g(X)$ is integrable. Then $\mathbb{E}[\mathrm{\mid }g(X)\mathrm{\mid }]<\mathrm{\infty }$, so step 2.1 applied to $\mathrm{\mid }g\mathrm{\mid }$ gives ${\int }_S\mathrm{\mid }g\mathrm{\mid }d{\mathbb{P}}_X=\mathbb{E}[\mathrm{\mid }g(X)\mathrm{\mid }]<\mathrm{\infty }\mathrm{.}$ Hence $g$ is ${\mathbb{P}}_X$-integrable. For real-valued $g$, [L4] and [L5] give $g=g^{+}-g^{-}$ with both parts nonnegative, so step 2.1 and linearity yield $\mathbb{E}[g(X)]=\mathbb{E}[g^{+}(X)]-\mathbb{E}[g^{-}(X)]={\int }_S{g}^{+}d{\mathbb{P}}_X-{\int }_S{g}^{-}d{\mathbb{P}}_X={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

If $g$ is complex-valued, write $g=u+iv$ with real measurable parts $u,v$. The inequality $\mathrm{\mid }u\mathrm{\mid },\mathrm{\mid }v\mathrm{\mid }\le \mathrm{\mid }g\mathrm{\mid }$ and step 3.1 show that $u(X)$ and $v(X)$ are integrable, so the real-valued case applied to $u$ and $v$, followed by complex-linearity from [L5], gives $\mathbb{E}[g(X)]=\mathbb{E}[u(X)]+i\mathbb{E}[v(X)]={\int }_{S}ud{\mathbb{P}}_X+i{\int }_{S}vd{\mathbb{P}}_X={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Step 2.1 proves the nonnegative case, while steps 3.1 and 4.1 prove the integrable real and complex cases.