Tonelli's Theorem

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 $f: X \times Y \to \overline \R_{\ge 0}$ be a positive $\Sigma_X \otimes \Sigma_Y$-measurable function.

Then:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \int_X \map f {x, y} \map {\d \mu} x \map {\d \nu} y = \int_X \int_Y \map f {x, y} \map {\d \nu} y \map {\d \mu} x$

Corollary

Let $\sequence {a_{n, m} }_{\tuple {n, m} \in \N^2}$ be a doubly subscripted sequence of non-negative real numbers.

Then:

$\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty a_{n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty a_{n, m} }$

Proof

We rewrite the demand as:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$

where $f_x$ is the $x$-vertical section of $f$, and $f^y$ is the $y$-horizontal section of $f$.

From Horizontal Section of Measurable Function is Measurable, we have:

$f^y$ is $\Sigma_X$-measurable.

From Vertical Section of Measurable Function is Measurable, we have:

$f_x$ is $\Sigma_Y$-measurable.
$\ds y \mapsto \int_X f^y \rd \mu$ is $\Sigma_Y$-measurable
$\ds x \mapsto \int_Y f_x \rd \nu$ is $\Sigma_X$-measurable.

So, both:

$\ds \int_Y \paren {\int_X f^y \rd \mu} \rd \nu$

and:

$\ds \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$

are well-defined.

We first prove the case of:

$f = \chi_E$

where $E$ is a $\Sigma_X \otimes \Sigma_Y$-measurable set.

Lemma 1

Let $E \in \Sigma_X \otimes \Sigma_Y$.

Let:

$f = \chi_E$

where $\chi_E$ is the characteristic function of $E$.

Then:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$

$\Box$

Now consider the case of a positive simple $f$.

Lemma 2

Let $f : X \times Y \to \overline \R$ be a positive simple function.

Then:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$

$\Box$

Now take a general positive $\Sigma_X \otimes \Sigma_Y$-measurable function $f$.

We first show:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu$
there exists a increasing sequence of positive simple functions $\sequence {f_n}_{n \mathop \in \N}$ such that $f_n \to f$ pointwise.

From Simple Function is Measurable, we have:

for each $n \in \N$, the function $f_n$ is $\Sigma_X \otimes \Sigma_Y$-measurable.

From Horizontal Section of Measurable Function is Measurable, we have:

for each $n \in \N$, the function $\paren {f_n}^y$ is $\Sigma_X$-measurable, for each $y \in Y$.
the sequence $\sequence {\paren {f_n}^y}_{n \mathop \in \N}$ is increasing for each $y \in Y$.

Now, from Horizontal Section preserves Pointwise Limits of Sequences of Functions, we have:

$\paren {f_n}^y \to f^y$ pointwise for each $y \in Y$.

For each $n \in \N$ define a function $g_n : X \to \overline \R$ by:

$\ds \map {g_n} y = \int_X \paren {f_n}^y \rd \mu$

for each $y \in Y$.

Since $\sequence {\paren {f_n}^y}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma_X$-measurable functions, we can apply the Monotone Convergence Theorem to obtain:

$\ds \lim_{n \mathop \to \infty} \map {g_n} y = \int_X f^y \rd \nu$

From Integral of Positive Measurable Function is Monotone, we have:

$\sequence {g_n}_{n \mathop \in \N}$ is increasing.
$\sequence {g_n}_{n \mathop \in \N}$ is $\Sigma_Y$-measurable.

For each $n \in \N$, we have:

 $\ds \int_{X \times Y} f_n \map \rd {\mu \times \nu}$ $=$ $\ds \int_Y \paren {\int_X \paren {f_n}^y \rd \mu} \rd \nu$ from our work with simple functions $\ds$ $=$ $\ds \int_Y g_n \rd \nu$

Since $\sequence {f_n}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma_X \otimes \Sigma_Y$-measurable functions converging pointwise to $f$, we have:

$\ds \lim_{n \mathop \to \infty} \int_{X \times Y} f_n \map \rd {\mu \times \nu} = \int_{X \times Y} f \map \rd {\mu \times \nu}$

from the Monotone Convergence Theorem.

Since $\sequence {g_n}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma_X \otimes \Sigma_Y$-measurable functions with:

$\ds g_n \to \int_X f^y \rd \mu$ pointwise.

So:

$\ds \lim_{n \mathop \to \infty} \int_Y g_n \rd \nu = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu$

by the Monotone Convergence Theorem.

So we obtain:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu$

We now show:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$
there exists a increasing sequence of positive simple functions $\sequence {f_n}_{n \mathop \in \N}$ such that $f_n \to f$ pointwise.

From Simple Function is Measurable, we have:

for each $n \in \N$, the function $f_n$ is $\Sigma_X \otimes \Sigma_Y$-measurable.

From Vertical Section of Measurable Function is Measurable, we have:

for each $n \in \N$, the function $\paren {f_n}_x$ is $\Sigma_Y$-measurable, for each $x \in X$.
the sequence $\sequence {\paren {f_n}_x}_{n \mathop \in \N}$ is increasing for each $x \in X$.

Now, from Vertical Section preserves Pointwise Limits of Sequences of Functions, we have:

$\paren {f_n}_x \to f_x$ pointwise for each $x \in X$.

For each $n \in \N$ define a function $h_n : X \to \overline \R$ by:

$\ds \map {h_n} x = \int_Y \paren {f_n}_x \rd \nu$

for each $x \in X$.

Since $\sequence {\paren {f_n}_x}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma_Y$-measurable functions, we can apply the Monotone Convergence Theorem to obtain:

$\ds \lim_{n \mathop \to \infty} \map {h_n} x = \int_Y f_x \rd \nu$

From Integral of Positive Measurable Function is Monotone, we have:

$\sequence {h_n}_{n \mathop \in \N}$ is increasing.
$\sequence {h_n}_{n \mathop \in \N}$ is $\Sigma_X$-measurable.

For each $n \in \N$, we have:

 $\ds \int_{X \times Y} f_n \map \rd {\mu \times \nu}$ $=$ $\ds \int_X \paren {\int_Y \paren {f_n}_x \rd \nu} \rd \mu$ from our work with simple functions $\ds$ $=$ $\ds \int_X h_n \rd \mu$

Since $\sequence {f_n}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma_X \otimes \Sigma_Y$-measurable functions converging pointwise to $f$, we have:

$\ds \lim_{n \mathop \to \infty} \int_{X \times Y} f_n \map \rd {\mu \times \nu} = \int_{X \times Y} f \map \rd {\mu \times \nu}$

from the Monotone Convergence Theorem.

Since $\sequence {h_n}_{n \mathop \in \N}$ is an increasing sequence of $\Sigma_X \otimes \Sigma_Y$-measurable functions with:

$\ds h_n \to \int_X f^y \rd \mu$ pointwise.

So:

$\ds \lim_{n \mathop \to \infty} \int_Y h_n \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$

by the Monotone Convergence Theorem.

So we obtain:

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

giving:

$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \int_X \map f {x, y} \map {\d \mu} x \map {\d \nu} y = \int_X \int_Y \map f {x, y} \map {\d \nu} y \map {\d \mu} x$

as was the demand.

$\blacksquare$

Source of Name

This entry was named for Leonida Tonelli.