Horizontal Section of Simple Function is Simple Function

Theorem

Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.

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

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

Let $y \in Y$.

Then $f^y : X \to \R$ is a simple function, where $f^y$ is the $y$-horizontal 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

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

We have:

\(\ds f^y\)

\(=\)

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

\(\ds \)

\(=\)

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

Horizontal Section of Linear Combination of Functions is Linear Combination of Horizontal Sections

\(\ds \)

\(=\)

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

Horizontal Section of Characteristic Function is Characteristic Function of Horizontal Section

From Intersection of Horizontal Sections is Horizontal Section of Intersection, we have that:

$\paren {E_1}^y, \paren {E_2}^y, \ldots, \paren {E_n}^y$ are pairwise disjoint.

From Horizontal Section of Measurable Set is Measurable, we have that:

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

So, we have:

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

with:

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

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

So $f^y$ is a simple function.

$\blacksquare$