Null Sets of a Product Measure, and Exchanging the Factors

In the setting of Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be measure spaces with $\mu$ and $\nu$ $\sigma$-finite.

(i) Sets of measure zero. For every $E\in\mathcal{F}\otimes\mathcal{G}$,

$$(\mu\otimes\nu)(E)=0\iff\nu(E_{x})=0\text{ for }\mu\text{-almost every }x\in X\iff\mu(E^{y})=0\text{ for }\nu\text{-almost every }y\in Y.$$

(ii) Exchanging the factors. Let $s:Y\times X\to X\times Y$ be the function $s(y,x)=(x,y)$, which 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 two factors exchanged). Then

$$(\nu\otimes\mu)\circ s^{-1}=\mu\otimes\nu.$$