# 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)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)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 $\displaystyle{f(x) = \sum_{k=1}^{n} a_k \chi_{E_k}(x)}$. |