Category:Fubini's Theorem

This category contains pages concerning Fubini's Theorem:

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

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

Let $f: X \times Y \to \overline \R$ 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:

$I_f$ is $\mu$-integrable and $J_f$ is $\nu$-integrable

and:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_X I_f \rd \mu = \int_Y J_f \rd \nu$