Iterated Integrals
Why
We integrate over a product space by integrating one coordinate at a time.
Result
Suppose $(X, \mathcal{A} , \mu )$ and $(Y,
\mathcal{A} , \nu )$ are $\sigma $-finite
measurable spaces.
Let $f: X \times Y \to \Rbar$ be
$\mathcal{A} \times \mathcal{B} $-measurable and
$\mu \times \nu $-integrable.
Then
- For $\mu $-almost every $x$ in $X$ the section $f_x$ is $\nu $-integrable and for $\nu $-almost every $y$ in $Y$ the section $f^y$ is $\mu $-integrable,
- the functions $I_f$ and $J_f$ defined by
\[ I_f(x) = \begin{cases} \int _{Y} f_x d\nu &\text{ if } f_x \text{ is } \nu \text{-integrable}, \\ 0 & \text{ otherwise} \end{cases} \]
and\[ J_f(y) = \begin{cases} \int _{X} f^y d\mu &\text{ if } f^y \text{ is } \mu \text{-integrable}, \\ 0 & \text{ otherwise} \end{cases} \]
belong to $\mathcal{L} (X, \mathcal{A} , \mu , \R )$ and $\mathcal{L} (Y, \mathcal{A} , \nu , \R )$ respectively, and - the relation
\[ \int _{X \times Y} f d(\mu \times \nu ) = \int _{X} I_f d\mu = \int _{Y} J_f d\nu \]
holds.
The above is called Fubini's Theorem. Next: Tonelli's theorem.