Proof of Independent Random Elements: Product Laws and Iterated Expectations

Throughout, $P_{X}$ and $P_{Y}$ are probability measures on $(S,\mathcal{B}(S))$ and $(T,\mathcal{B}(T))$, hence $\sigma$-finite, the constant sequence of the whole space witnessing it, so that $P_{X}\otimes P_{Y}$ and Tonelli's and Fubini's theorems are available.

Step 1: the joint law. The function $(X,Y):\Omega\to S\times T$ is $\mathcal{F}/(\mathcal{B}(S)\otimes\mathcal{B}(T))$-measurable (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (vii)), so $\{(X,Y)\in C\}=(X,Y)^{-1}(C)$ is an event for every $C\in\mathcal{B}(S)\otimes\mathcal{B}(T)$, and $Q=P\circ(X,Y)^{-1}$ is a probability measure on $(S\times T,\mathcal{B}(S)\otimes\mathcal{B}(T))$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)). For $A\in\mathcal{B}(S)$ and $B\in\mathcal{B}(T)$ we have $(X,Y)^{-1}(A\times B)=X^{-1}(A)\cap Y^{-1}(B)$, so by The Law of a Random Element (i) and Product Measure (i)

$$Q(A\times B)=P\bigl(X^{-1}(A)\cap Y^{-1}(B)\bigr),\qquad(P_{X}\otimes P_{Y})(A\times B)=P\bigl(X^{-1}(A)\bigr)\,P\bigl(Y^{-1}(B)\bigr).\tag{1}$$

Step 2: independence on rectangles. As $\{X\in A,\ Y\in B\}=X^{-1}(A)\cap Y^{-1}(B)$, $\{X\in A\}=X^{-1}(A)$ and $\{Y\in B\}=Y^{-1}(B)$ (Probability Spaces: Notation (iii)), Independent Random Elements: Preimages, Functions and Blocks (ii) shows that $X$ and $Y$ are independent if and only if

$$P\bigl(X^{-1}(A)\cap Y^{-1}(B)\bigr)=P\bigl(X^{-1}(A)\bigr)\,P\bigl(Y^{-1}(B)\bigr)\qquad(A\in\mathcal{B}(S),\ B\in\mathcal{B}(T)).\tag{2}$$

Proof of (i). By (1), condition (2) says that $Q$ and $P_{X}\otimes P_{Y}$ agree on the measurable rectangles. If it holds, the two measures agree on $\mathrm{Rect}(\mathcal{B}(S),\mathcal{B}(T))$, a $\pi$-system containing $S\times T$ (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (i)) that generates $\mathcal{B}(S)\otimes\mathcal{B}(T)$ (Measurable Rectangles and the Product σ-Algebra (ii)); both give the whole space the finite mass $1$, so $Q=P_{X}\otimes P_{Y}$ by Uniqueness of Measures Agreeing on a Generating π-System (i), which is the displayed identity. Conversely, the displayed identity for $C=A\times B$ is (2) by (1).

Proof of (ii). As $X$ and $Y$ are independent, $Q=P_{X}\otimes P_{Y}$ by (i); and $g(X,Y)=g\circ(X,Y)$. Change of variables (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (ii)) gives $g\circ(X,Y)\in\mathcal{M}^{+}(\Omega,\mathcal{F})$, and with Tonelli's theorem (Tonelli's and Fubini's Theorems (i))

$$\mathbb{E}\,g(X,Y)=\int_{\Omega}g\circ(X,Y)\,dP=\int_{S\times T}g\,d(P_{X}\otimes P_{Y})=\int_{S}\Bigl(\int_{T}g_{x}\,dP_{Y}\Bigr)dP_{X}(x).$$

Proof of (iii). Again $Q=P_{X}\otimes P_{Y}$. By change of variables (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (iii)), $g(X,Y)=g\circ(X,Y)$ is a random variable, it lies in $\mathcal{L}^{1}(P)$ if and only if $g\in\mathcal{L}^{1}(P_{X}\otimes P_{Y})$, and then $\mathbb{E}\,g(X,Y)=\int_{S\times T}g\,d(P_{X}\otimes P_{Y})$. Fubini's theorem (Tonelli's and Fubini's Theorems (ii)) gives the assertions on $N$ and $I$ and the equality of this integral with $\int_{S}I\,dP_{X}$.