Measures of Sections of a Product-Measurable Set

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be measure spaces with $\nu$ $\sigma$-finite. Let $\mathcal{F}\otimes\mathcal{G}$ be the product $\sigma$-algebra on $X\times Y$, and for $E\subseteq X\times Y$ and $x\in X$ let $E_{x}$ be the section of $E$ at $x$.

(i) Measurability. For every $E\in\mathcal{F}\otimes\mathcal{G}$, the function $x\mapsto\nu(E_{x})$ lies in $\mathcal{M}^{+}(X,\mathcal{F})$ (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (iv)).

(ii) A measure on the product. The function $\kappa:\mathcal{F}\otimes\mathcal{G}\to[0,\infty]$ given by

$$\kappa(E)=\int_{X}\nu(E_{x})\,d\mu(x)$$

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

(iii) Rectangles. For all $A\in\mathcal{F}$ and $B\in\mathcal{G}$,

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