Let $(X, \mathcal{A} , \mu )$ and $(Y, \mathcal{B} , \nu )$ be $\sigma $-finite measurable spaces. Let $E \in \mathcal{A} \times \mathcal{B} $. Let $\mu = \mu _1 \times \mu _2$ be the product measure on the product sigma algebra $\mathcal{A} _1 \times \mathcal{A} _2$. Let $f: X_1 \times X_2 \to \eri$ be the indicator of $E$. Then: TODO