Existence and Uniqueness of the Product Measure

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 $\mu$ and $\nu$ $\sigma$-finite. Let $\mathcal{F}\otimes\mathcal{G}$ be the product $\sigma$-algebra on $X\times Y$, and for $E\subseteq X\times Y$, $x\in X$ and $y\in Y$ let $E_{x}$ and $E^{y}$ be the sections of $E$ at $x$ and at $y$.

(i) Existence and uniqueness. There is exactly one measure $\pi$ on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$ with

$$\pi(A\times B)=\mu(A)\,\nu(B)\qquad\text{for all }A\in\mathcal{F}\text{ and }B\in\mathcal{G}.$$

(ii) $\sigma$-finiteness. The measure $\pi$ of (i) is $\sigma$-finite.

(iii) Measures of sections. For every $E\in\mathcal{F}\otimes\mathcal{G}$, the functions $x\mapsto\nu(E_{x})$ and $y\mapsto\mu(E^{y})$ lie in $\mathcal{M}^{+}(X,\mathcal{F})$ and $\mathcal{M}^{+}(Y,\mathcal{G})$, and the measure $\pi$ of (i) satisfies

$$\pi(E)=\int_{X}\nu(E_{x})\,d\mu(x)=\int_{Y}\mu(E^{y})\,d\nu(y).$$