Fubini's Theorem/Lemma

Lemma
Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.

Let $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$.

Let $f: X \times Y \to \R$ be a $\mu \times \nu$-integrable function.

Define a function $I_f : X \to \R$ by:


 * $\ds \map {I_f} x = \begin{cases}\ds \int_Y f_x \rd \nu & \text {if } f_x \text { is } \nu\text{-integrable} \\ 0 & \text{otherwise}\end{cases}$

for each $x \in X$.

Define a function $J_f : Y \to \R$ by:


 * $\ds \map {J_f} y = \begin{cases}\ds \int_X f^y \rd \mu & \text {if } f^y \text { is } \mu\text{-integrable} \\ 0 & \text{otherwise}\end{cases}$

for each $y \in Y$.

Then:


 * $I_f$ and $J_f$ are $\Sigma$-measurable.