The Lebesgue Integral of Simple Functions

# 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)
\begin{align} \quad E_k = \{ x \in [a, b] : \varphi(x) = a_k \} \end{align}

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)
\begin{align} \quad \varphi(x) = \sum_{k=1}^{n} a_k \chi_{E_k}(x) \end{align}
 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)
\begin{align} \quad \varphi (x) = \left\{\begin{matrix} 1 & \mathrm{if} \: x \in \mathbb{Q} \cap [0, 1] \\ 0 & \mathrm{if} \: x \in [0, 1] \setminus \mathbb{Q} \end{matrix}\right. \end{align}

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)
\begin{align} \quad \varphi(x) = a_1 \chi_{E_1}(x) + a_2 \chi_{E_2}(x) \end{align}

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

(5)
\begin{align} \quad (L) \int_{[0, 1]} \varphi &= a_1 m(E_1) + a_2 m(E_2) \\ &= (1)(0) + (0)(1) \\ &= 0 \end{align}

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:
(6)
\begin{align} \quad F_j = \bigcup_{k \in \{ 1, 2, ..., n \}, \: a_k = b_j} E_k \end{align}
• Then we can write $\varphi(x)$ in canonical notation as:
(7)
\begin{align} \quad \varphi(x) = \sum_{j=1}^{m} b_j \chi_{E_j}(x) \end{align}
• Noting that the sets $E_1, E_2, ..., E_n$ are disjoint, we have that the integral of $\varphi$ on $E$ is:
(8)
\begin{align} \quad (L) \int_E \varphi &= \sum_{j=1}^{m} b_j m(E_j) \\ &= \sum_{j=1}^{m} b_j \left ( \sum_{k \in \{ 1, 2, ..., n \}, \: a_k = b_j} m(E_k) \right ) \\ &= \sum_{k=1}^{n} a_k m(E_k) \quad \blacksquare \end{align}