Lebesgue Measurable Simple Functions

Lebesgue Measurable Simple Functions

 Definition: A Lebesgue Measurable Simple Function $f$ is a Lebesgue measurable function whose range is a finite set.

We will often refer to Lebesgue measurable simple functions as just "Simple Functions".

Some graphs of various simple functions are given below. For example, every constant function defined on a Lebesgue measurable set is a simple function such as $f : \mathbb{R} \to \{ c \}$ defined for all $x \in \mathbb{R}$ by $f(x) = c$ where $c \in \mathbb{R}$.

Furthermore, if $f : D \to \mathbb{R}$ and $D$ is a finite set then $f$ is a simple function. This is because every finite subset of $\mathbb{R}$ is a Lebesgue measurable set (since $m(D) = 0$) and furthermore $| D(f) | \geq | R(f) |$ and so $R(f)$ is a finite set.

We now define a special type of simple function below.

 Definition: Let $E \subseteq F$. The Characteristic Function of $E$ on the set $F$ is the function $\chi_E : F \to \{ 0, 1 \}$ defined for all $x \in F$ by $\displaystyle{\chi_E (x) = \left\{\begin{matrix} 0 & \mathrm{if} \: x \in F \setminus E \\ 1 & \mathrm{if} \: x \in E \end{matrix}\right.}$.

Note that if $F$ is a Lebesgue measurable set then $\chi_E$ is a simple function.

The Canonical Representation of a Simple Function

Let $f$ be a simple function. Then $R(f) = \{ a_1, a_2, ..., a_n \}$ where $a_1, a_2, ..., a_n \in \mathbb{R}$ are all distinct. For each $k \in \{ 1, 2, ..., n \}$ let:

(1)
\begin{align} \quad E_k = \{ x \in D(f) : f(x) = a_k \} \end{align}

In other words, $E_k$ is the subset of the domain of $f$ containing all points that map to $a_k$ in the range of $f$. Let $\chi_{E_1}$, $\chi_{E_2}$, …, $\chi_{E_n}$ denote the characteristic functions of $E_1$, $E_2$, …, $E_n$ on $D(f)$. Then $f$ can be expressed as a linear combination of these functions, namely:

(2)
\begin{align} \quad f(x) &= \sum_{k=1}^{n} a_k \chi_{E_k}(x) \\ &= a_1\chi_{E_1}(x) + a_2\chi_{E_2}(x) + ... + a_n\chi_{E_n}(x) \end{align}

This representation of a simple function in terms of characteristic functions is given a special name.

 Definition: If $f$ is a simple function with range $R(f) = \{ a_1, a_2, ..., a_n \}$ and $E_k = \{ x \in D(f) : f(x) = a_k \}$ for $k \in \{ 1, 2, ..., n \}$ then the Canonical Representation of $f$ is $\displaystyle{f(x) = \sum_{k=1}^{n} a_k \chi_{E_k}(x)}$.