Tonelli's Theorem/Lemma 2

Lemma

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 positive simple function.

Then:

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

where:

$f^y$ is the $y$-horizontal section of $f$

$f_x$ is the $x$-vertical section of $f$.

Proof

Write the standard representation of $f$ as:

$\ds f = \sum_{k \mathop = 1}^n a_k \chi_{E_k}$

with:

$E_1, E_2, \ldots, E_n$ pairwise disjoint $\Sigma_X \otimes \Sigma_Y$-measurable sets

$a_1, a_2, \ldots, a_n$ non-negative real numbers.

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

$f^y$ is a positive simple function

with:

$\ds f^y = \sum_{k \mathop = 1}^n a_k \chi_{\paren {E_k}^y}$

where:

$\paren {E_1}^y, \paren {E_2}^y, \ldots, \paren {E_n}^y$ are pairwise disjoint $\Sigma_X$-measurable sets

$a_1, a_2, \ldots, a_n$ are real numbers.

From the definition of the $\mu$-integral of a positive simple function, we have:

$\ds \map {I_\mu} {f^y} = \sum_{k \mathop = 1}^n a_k \map \mu {\paren {E_k}^y}$

From Integral of Positive Measurable Function Extends Integral of Positive Simple Function, we then have:

$\ds \int f^y \rd \mu = \sum_{k \mathop = 1}^n a_k \map \mu {\paren {E_k}^y}$

Similarly, from Vertical Section of Simple Function is Simple Function, we have:

$f_x$ is a positive simple function

with:

$\ds f_x = \sum_{k \mathop = 1}^n a_k \chi_{\paren {E_k}_x}$

where:

$\paren {E_1}_x, \paren {E_2}_x, \ldots, \paren {E_n}_x$ are pairwise disjoint $\Sigma_X$-measurable sets

$a_1, a_2, \ldots, a_n$ are non-negative real numbers.

From the definition of the $\nu$-integral of a positive simple function, we have:

$\ds \map {I_\nu} {f_x} = \sum_{k \mathop = 1}^n a_k \map \nu {\paren {E_k}_x}$

From Integral of Positive Measurable Function Extends Integral of Positive Simple Function, we then have:

$\ds \int f_x \rd \nu = \sum_{k \mathop = 1}^n a_k \map \nu {\paren {E_k}_x}$

We then have:

\(\ds \int_{X \times Y} f \map \rd {\mu \times \nu}\)

\(=\)

\(\ds \int_{X \times Y} \paren {\sum_{k \mathop = 1}^n a_k \chi_{E_k} } \map \rd {\mu \times \nu}\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \paren {\int_{X \times Y} a_k \chi_{E_k} \map \rd {\mu \times \nu} }\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \paren {\int_{X \times Y} a_k \chi_{\paren {E_k}^y} \map \rd {\mu \times \nu} }\)

Integral of Positive Measurable Function is Additive

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \paren {\int_{X \times Y} \chi_{E_k} \map \rd {\mu \times \nu} }\)

Integral of Positive Measurable Function is Positive Homogeneous

With a view to show:

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

write:

\(\ds \sum_{k \mathop = 1}^n a_k \paren {\int_{X \times Y} \chi_{E_k} \map \rd {\mu \times \nu} }\)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \int_Y \paren {\int_X \paren {\chi_{E_k} }^y \rd \mu} \rd \nu\)

from our work on characteristic functions

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \int_Y \paren {\int_X \chi_{\paren {E_k}^y} \rd \mu} \rd \nu\)

Horizontal Section of Characteristic Function is Characteristic Function of Horizontal Section

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \int_Y \paren {\map \mu {\paren {E_k}^y} } \rd \nu\)

Integral of Characteristic Function: Corollary

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \paren {\int_Y \paren {a_k \map \mu {\paren {E_k}^y} } \rd \nu}\)

Integral of Positive Measurable Function is Positive Homogeneous

\(\ds \)

\(=\)

\(\ds \int_Y \paren {\sum_{k \mathop = 1}^n a_k \map \mu {\paren {E_k}^y} } \rd \nu\)

Integral of Positive Measurable Function is Additive

\(\ds \)

\(=\)

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

To show that:

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

write:

\(\ds \sum_{k \mathop = 1}^n a_k \paren {\int_{X \times Y} \chi_{E_k} \map \rd {\mu \times \nu} }\)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \int_X \paren {\int_Y \paren {\chi_{E_k} }_x \rd \nu} \rd \mu\)

from our work on characteristic functions

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \int_X \paren {\int_Y \chi_{\paren {E_k}_x} \rd \nu} \rd \mu\)

Vertical Section of Characteristic Function is Characteristic Function of Vertical Section

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n a_k \int_X \paren {\map \nu {\paren {E_k}_x} } \rd \mu\)

Integral of Characteristic Function: Corollary

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \paren {\int_X \paren {a_k \map \nu {\paren {E_k}_x} } \rd \mu}\)

Integral of Positive Measurable Function is Positive Homogeneous

\(\ds \)

\(=\)

\(\ds \int_X \paren {\sum_{k \mathop = 1}^n a_k \map \nu {\paren {E_k}_x} } \rd \mu\)

Integral of Positive Measurable Function is Additive

\(\ds \)

\(=\)

\(\ds \int_X \paren {\int_Y f_x \rd \nu} \rd \mu\)

So:

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

$\blacksquare$