# The Lebesgue Integral of Simple Functions

Recall from the Lebesgue Measurable Simple Functions page that a simple function is a Lebesgue measurable function whose range is a finite set.

Let $\varphi$ be a simple function defined on a Lebesgue measurable set $E$ with $m(E) < \infty$. Then $\varphi$ takes on only finitely many values, say $a_1, a_2, ..., a_n$. Let $E_1, E_2, ..., E_n \subseteq E$ be defined for each $k \in \{ 1, 2, ..., n \}$ by:

(1)We let $\displaystyle{\chi_{E_k} = \left\{\begin{matrix} 0 & \mathrm{if} \: x \in E \setminus E_k \\ 1 & \mathrm{if} \: x \in E_k \end{matrix}\right.}$ denote the characteristic function on $E_k$ to $E$. Then we can write $\varphi (x)$ in its canonical representation as:

(2)Definition: The Lebesgue Integral of the Simple Function $\varphi$ defined on a Lebesgue measurable set $E$ with $m(E) < \infty$ and with canonical representation $\displaystyle{ \varphi(x) = \sum_{k=1}^{n} a_k \chi_{E_k}}$ is $\displaystyle{(L) \int_E \varphi (x) \: dx = \sum_{k=1}^{n} a_k m(E_k)}$. |

*The added "$(L)$" in the notation for the Lebesgue integral of $\varphi$ above is omitted when no ambiguity arises. We will also sometimes use the shorter notation "$\displaystyle{(L) \int_E \varphi}$" or "$\displaystyle{\int_E \varphi}$" to denote the integral above.*

For example, consider the following function $\varphi$ defined on the Lebesgue measurable set $E = [0, 1]$:

(3)Clearly $\varphi$ is a simple function with range $R(\varphi) = \{ 0, 1 \}$. We let $E_1 = \mathbb{Q} \cap [0, 1]$ and let $E_2 = [0, 1] \setminus \mathbb{Q}$, and let $a_1 = 1$, $a_2 = 0$. The canonical representation for $\varphi(x)$ is:

(4)And the Lebesgue integral of $\varphi$ on $[0, 1]$ is:

(5)The following proposition gives us a less restrictive way to represent the Lebesgue integral of a simple function.

Proposition 1: Let $\varphi$ be a simple function defined on a Lebesgue measurable set $E$ with $m(E) < \infty$. If $E_1, E_2, ..., E_n$ are any mutually disjoint subsets of $E$ and such that $\displaystyle{\varphi(x) = \sum_{k=1}^{n} a_k \chi_{E_k}(x)}$ then $\displaystyle{(L) \int_E \varphi = \sum_{k=1}^{n} a_k m(E_k)}$. |

**Proof:**Let $\varphi$ be a simple function defined on $E$ with $m(E) < \infty$ and let $\{ a_1, a_2, ..., a_n \} = \{ b_1, b_2, ..., b_m \}$ where the $b_j$s are each distinct. For each $j \in \{ 1, 2, ..., m \}$, let $F_j$ denote the union of the sets $E_k$ for which $a_k = b_j$, that is:

- Then we can write $\varphi(x)$ in canonical notation as:

- Noting that the sets $E_1, E_2, ..., E_n$ are disjoint, we have that the integral of $\varphi$ on $E$ is: