Proof of Measures of Sections of a Product-Measurable Set

Throughout, $E_{x}\in\mathcal{G}$ for $E\in\mathcal{F}\otimes\mathcal{G}$ and $x\in X$ (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (iv)), and sections commute with the set operations (Rectangles, Sections and the Swap: Set Identities (ii), Rectangles, Sections and the Swap: Set Identities (i)): for $E\subseteq X\times Y$, a sequence $(E_{n})$ of subsets of $X\times Y$, $A\in\mathcal{F}$, $B\in\mathcal{G}$ and $x\in X$,

$$\bigl((X\times Y)\setminus E\bigr)_{x}=Y\setminus E_{x},\qquad\Bigl(\bigcup_{n\in\mathbb{N}}E_{n}\Bigr)_{x}=\bigcup_{n\in\mathbb{N}}(E_{n})_{x},\qquad(A\times B)_{x}=\begin{cases}B,&x\in A,\\\emptyset,&x\notin A,\end{cases}$$

and disjoint subsets of $X\times Y$ have disjoint sections.

Step 1: a set of finite measure. Fix $D\in\mathcal{G}$ with $\nu(D)<\infty$. For $E\in\mathcal{F}\otimes\mathcal{G}$ let $g_{E}:X\to[0,\infty]$ be the function

$$g_{E}(x)=\nu(E_{x}\cap D);$$

its values are real, as $g_{E}(x)\le\nu(D)<\infty$ (Basic Properties of a Measure (iii)). Let $\mathcal{L}$ be the set of those $E\in\mathcal{F}\otimes\mathcal{G}$ with $g_{E}\in\mathcal{M}^{+}(X,\mathcal{F})$. We show that $\mathcal{L}$ is a $\lambda$-system on $X\times Y$ containing the set $\mathrm{Rect}(\mathcal{F},\mathcal{G})$ of measurable rectangles.

Rectangles. Let $A\in\mathcal{F}$ and $B\in\mathcal{G}$. By the section of $A\times B$ displayed above, and as $\nu(\emptyset)=0$,

$$g_{A\times B}(x)=\nu(B\cap D)\,\mathbf{1}_{A}(x)\qquad(x\in X).$$

Since $\nu(B\cap D)$ is a nonnegative real number and $\mathbf{1}_{A}\in\mathcal{M}^{+}(X,\mathcal{F})$ with $\int_{X}\mathbf{1}_{A}\,d\mu=\mu(A)$ (Basic Properties of the Integral of Nonnegative Measurable Maps (ii)), Basic Properties of the Integral of Nonnegative Measurable Maps (iv) gives $g_{A\times B}\in\mathcal{M}^{+}(X,\mathcal{F})$ and

$$\int_{X}g_{A\times B}\,d\mu=\nu(B\cap D)\,\mu(A).\tag{1}$$

Thus $\mathrm{Rect}(\mathcal{F},\mathcal{G})\subseteq\mathcal{L}$; in particular $X\times Y\in\mathcal{L}$, as $X\times Y\in\mathrm{Rect}(\mathcal{F},\mathcal{G})$ (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (i)).

Complements. Let $E\in\mathcal{L}$ and $E'=(X\times Y)\setminus E$, which lies in $\mathcal{F}\otimes\mathcal{G}$. For $x\in X$ we have $E'_{x}\cap D=D\setminus(E_{x}\cap D)$, so by Basic Properties of a Measure (iv)

$$g_{E'}(x)=\nu(D)-g_{E}(x),$$

a difference in $\mathbb{R}$. As a real-valued function, $g_{E}$ is measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)), and so is the constant function $\nu(D)$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)); hence their difference $g_{E'}$ is measurable (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i)), and since $g_{E'}\ge0$, it lies in $\mathcal{M}^{+}(X,\mathcal{F})$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)). So $E'\in\mathcal{L}$.

Disjoint unions. Let $(E_{n})$ be a pairwise disjoint sequence in $\mathcal{L}$ and $E=\bigcup_{n\in\mathbb{N}}E_{n}$, which lies in $\mathcal{F}\otimes\mathcal{G}$. For $x\in X$ the sets $(E_{n})_{x}\cap D$ are pairwise disjoint with union $E_{x}\cap D$, so countable additivity of $\nu$ (Measure, Measure Space, and Probability Measure (i)) gives

$$g_{E}(x)=\sum_{n\in\mathbb{N}}g_{E_{n}}(x)\qquad(x\in X),$$

and $g_{E}\in\mathcal{M}^{+}(X,\mathcal{F})$ by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (ii). So $E\in\mathcal{L}$.

As $\mathcal{L}\subseteq\mathcal{F}\otimes\mathcal{G}\subseteq\mathcal{P}(X\times Y)$, this shows that $\mathcal{L}$ is a $\lambda$-system on $X\times Y$. Since $\mathrm{Rect}(\mathcal{F},\mathcal{G})$ is a $\pi$-system (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (i)) contained in $\mathcal{L}$, the $\pi$-$\lambda$ theorem (Dynkin's Pi-Lambda Theorem (ii)) gives $\mathcal{F}\otimes\mathcal{G}=\sigma(\mathrm{Rect}(\mathcal{F},\mathcal{G}))\subseteq\mathcal{L}$ (Measurable Rectangles and the Product σ-Algebra (ii)). Hence $g_{E}\in\mathcal{M}^{+}(X,\mathcal{F})$ for every $E\in\mathcal{F}\otimes\mathcal{G}$.

Step 2: a disjoint cover of $Y$ by sets of finite measure. Since $\nu$ is $\sigma$-finite, fix a sequence $(Y_{k})$ in $\mathcal{G}$ with $\bigcup_{k\in\mathbb{N}}Y_{k}=Y$ and $\nu(Y_{k})<\infty$ for every $k\in\mathbb{N}$ (Measure, Measure Space, and Probability Measure (iv)), and let $(D_{k})$ be its disjointing sequence, the sequence $(B_{n})$ of The Disjointing Sequence of a Sequence of Sets. Then $D_{k}\in\mathcal{G}$ (The Disjointing Sequence of a Sequence of Sets (v)), the $D_{k}$ are pairwise disjoint (The Disjointing Sequence of a Sequence of Sets (iii)) with union $Y$ (The Disjointing Sequence of a Sequence of Sets (iv)), and $D_{k}\subseteq Y_{k}$ (The Disjointing Sequence of a Sequence of Sets (ii)), so $\nu(D_{k})\le\nu(Y_{k})<\infty$. For every $C\in\mathcal{G}$ the sets $C\cap D_{k}$ are pairwise disjoint with union $C$ (Monotone Sequences of Sets, Unions of Traces, and Disjointness of Smaller Sets (iii), Monotone Sequences of Sets, Unions of Traces, and Disjointness of Smaller Sets (v)), so countable additivity gives

$$\nu(C)=\sum_{k\in\mathbb{N}}\nu(C\cap D_{k}).\tag{2}$$

Step 3: proof of (i). Let $E\in\mathcal{F}\otimes\mathcal{G}$. By (2) with $C=E_{x}$,

$$\nu(E_{x})=\sum_{k\in\mathbb{N}}\nu(E_{x}\cap D_{k})\qquad(x\in X).$$

By Step 1 with $D=D_{k}$, each function $x\mapsto\nu(E_{x}\cap D_{k})$ lies in $\mathcal{M}^{+}(X,\mathcal{F})$, so by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (ii) the function $x\mapsto\nu(E_{x})$ lies in $\mathcal{M}^{+}(X,\mathcal{F})$, and

$$\kappa(E)=\sum_{k\in\mathbb{N}}\int_{X}\nu(E_{x}\cap D_{k})\,d\mu(x).\tag{3}$$

Step 4: proof of (ii). By (i), $\kappa$ is a function from $\mathcal{F}\otimes\mathcal{G}$ to $[0,\infty]$. As $\emptyset_{x}=\emptyset$ for every $x$, $\kappa(\emptyset)=\int_{X}0\,d\mu=0$ (Basic Properties of the Integral of Nonnegative Measurable Maps (v)). Let $(E_{n})$ be a pairwise disjoint sequence in $\mathcal{F}\otimes\mathcal{G}$ with union $E$. For $x\in X$ the sections $(E_{n})_{x}$ are pairwise disjoint with union $E_{x}$, so $\nu(E_{x})=\sum_{n\in\mathbb{N}}\nu((E_{n})_{x})$, and by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (ii)

$$\kappa(E)=\int_{X}\sum_{n\in\mathbb{N}}\nu\bigl((E_{n})_{x}\bigr)\,d\mu(x)=\sum_{n\in\mathbb{N}}\int_{X}\nu\bigl((E_{n})_{x}\bigr)\,d\mu(x)=\sum_{n\in\mathbb{N}}\kappa(E_{n}).$$

So $\kappa$ is a measure on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$.

Step 5: proof of (iii). Let $A\in\mathcal{F}$ and $B\in\mathcal{G}$. By (3), then (1) with $D=D_{k}$, then Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence (ix) and (2) with $C=B$,

$$\kappa(A\times B)=\sum_{k\in\mathbb{N}}\nu(B\cap D_{k})\,\mu(A)=\mu(A)\sum_{k\in\mathbb{N}}\nu(B\cap D_{k})=\mu(A)\,\nu(B).$$