Proof of Existence and Uniqueness of the Product Measure

Let $\kappa$ be the measure on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$ of Measures of Sections of a Product-Measurable Set (ii), $\kappa(E)=\int_{X}\nu(E_{x})\,d\mu(x)$; it is defined since $\nu$ is $\sigma$-finite.

Existence. By Measures of Sections of a Product-Measurable Set (iii), $\kappa(A\times B)=\mu(A)\,\nu(B)$ for all $A\in\mathcal{F}$ and $B\in\mathcal{G}$.

A cover by rectangles of finite measure. Since $\mu$ and $\nu$ are $\sigma$-finite, fix sequences $(X_{i})$ in $\mathcal{F}$ and $(Y_{j})$ in $\mathcal{G}$ with $\bigcup_{i\in\mathbb{N}}X_{i}=X$, $\bigcup_{j\in\mathbb{N}}Y_{j}=Y$, and $\mu(X_{i})<\infty$, $\nu(Y_{j})<\infty$ for all $i,j$ (Measure, Measure Space, and Probability Measure (iv)), and fix a bijection $\beta:\mathbb{N}\to\mathbb{N}\times\mathbb{N}$ (Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ii)), $\beta(k)=(i_{k},j_{k})$. Put

$$R_{k}=X_{i_{k}}\times Y_{j_{k}}\qquad(k\in\mathbb{N}),$$

a measurable rectangle, so $R_{k}\in\mathrm{Rect}(\mathcal{F},\mathcal{G})\subseteq\mathcal{F}\otimes\mathcal{G}$ (Measurable Rectangles and the Product σ-Algebra (i), Measurable Rectangles and the Product σ-Algebra (ii), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)). The $R_{k}$ have union $X\times Y$: each $R_{k}$ is a subset of $X\times Y$, as $X_{i_{k}}\subseteq X$ and $Y_{j_{k}}\subseteq Y$; and for $(x,y)\in X\times Y$ pick $i,j$ with $x\in X_{i}$ and $y\in Y_{j}$, so that $(x,y)\in R_{k}$ for $k=\beta^{-1}(i,j)$. For every measure $\pi$ with the rectangle property of (i),

$$\pi(R_{k})=\mu(X_{i_{k}})\,\nu(Y_{j_{k}})<\infty,$$

since $\mu(X_{i_{k}})$ and $\nu(Y_{j_{k}})$ lie in $\mathbb{R}$, so that their product is the product in $\mathbb{R}$ (Addition and Multiplication on [0,∞] (ii)), which is finite.

Uniqueness. Let $\pi$ and $\pi'$ be measures on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$ with the rectangle property of (i). They agree on $\mathrm{Rect}(\mathcal{F},\mathcal{G})$, which is a $\pi$-system (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (i)) generating $\mathcal{F}\otimes\mathcal{G}$ (Measurable Rectangles and the Product σ-Algebra (ii)), and $(R_{k})$ is a sequence in $\mathrm{Rect}(\mathcal{F},\mathcal{G})$ with union $X\times Y$ and $\pi(R_{k})<\infty$. By Uniqueness of Measures Agreeing on a Generating π-System (ii), $\pi=\pi'$. With the existence of $\kappa$ this proves (i); from now on $\pi$ is the measure of (i), and $\pi=\kappa$.

Proof of (ii). $(R_{k})$ is a sequence in $\mathcal{F}\otimes\mathcal{G}$ with union $X\times Y$ and $\pi(R_{k})<\infty$ for every $k$, so $\pi$ is $\sigma$-finite (Measure, Measure Space, and Probability Measure (iv)).

Proof of (iii): the first formula. By Measures of Sections of a Product-Measurable Set (i), $x\mapsto\nu(E_{x})$ lies in $\mathcal{M}^{+}(X,\mathcal{F})$, and $\pi(E)=\kappa(E)=\int_{X}\nu(E_{x})\,d\mu(x)$.

Proof of (iii): the second formula. Let $t:Y\times X\to X\times Y$ be the swap $t(y,x)=(x,y)$; it is $(\mathcal{G}\otimes\mathcal{F})/(\mathcal{F}\otimes\mathcal{G})$-measurable (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (viii), with the factors $(Y,\mathcal{G})$ and $(X,\mathcal{F})$), so $t^{-1}(E)\in\mathcal{G}\otimes\mathcal{F}$ for every $E\in\mathcal{F}\otimes\mathcal{G}$. For $E\subseteq X\times Y$, $y\in Y$, $A\subseteq X$ and $B\subseteq Y$ (Rectangles, Sections and the Swap: Set Identities (iv)),

$$\bigl(t^{-1}(E)\bigr)_{y}=E^{y},\qquad t^{-1}(A\times B)=B\times A .$$

For the measure spaces $(Y,\mathcal{G},\nu)$ and $(X,\mathcal{F},\mu)$, taken in this order, the second measure $\mu$ is $\sigma$-finite. So, for every $E'\in\mathcal{G}\otimes\mathcal{F}$, the function $y\mapsto\mu(E'_{y})$ lies in $\mathcal{M}^{+}(Y,\mathcal{G})$ (Measures of Sections of a Product-Measurable Set (i)), and there is a measure $\kappa'$ on $(Y\times X,\mathcal{G}\otimes\mathcal{F})$ with

$$\kappa'(E')=\int_{Y}\mu(E'_{y})\,d\nu(y),\qquad\kappa'(B\times A)=\nu(B)\,\mu(A)$$

for $E'\in\mathcal{G}\otimes\mathcal{F}$, $B\in\mathcal{G}$, $A\in\mathcal{F}$ (Measures of Sections of a Product-Measurable Set (ii), Measures of Sections of a Product-Measurable Set (iii)). With $E'=t^{-1}(E)$, the first assertion shows that $y\mapsto\mu(E^{y})$ lies in $\mathcal{M}^{+}(Y,\mathcal{G})$ for every $E\in\mathcal{F}\otimes\mathcal{G}$.

Let $\pi'=\kappa'\circ t^{-1}$ be the image measure of $\kappa'$ under $t$, the function $E\mapsto\kappa'(t^{-1}(E))$ on $\mathcal{F}\otimes\mathcal{G}$; it is a measure on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)). For $A\in\mathcal{F}$ and $B\in\mathcal{G}$,

$$\pi'(A\times B)=\kappa'(B\times A)=\nu(B)\,\mu(A)=\mu(A)\,\nu(B).$$

By the uniqueness in (i), $\pi'=\pi$, so for every $E\in\mathcal{F}\otimes\mathcal{G}$

$$\pi(E)=\kappa'\bigl(t^{-1}(E)\bigr)=\int_{Y}\mu\bigl((t^{-1}(E))_{y}\bigr)\,d\nu(y)=\int_{Y}\mu(E^{y})\,d\nu(y).$$