Vertical 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 $x \in X$.

Then $f_x : Y \to \R$ is a simple function, where $f_x$ 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

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

We have:

\(\ds f_x\)

\(=\)

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

\(\ds \)

\(=\)

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

Vertical Section of Linear Combination of Functions is Linear Combination of Vertical Sections

\(\ds \)

\(=\)

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

Vertical Section of Characteristic Function is Characteristic Function of Vertical Section

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

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

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

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

So, we have:

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

with:

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

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

So $f_x$ is a simple function.

$\blacksquare$