Simple Integrals

Why

We want to define area under a real function. We begin with functions whose area under the curve is self-evident.

Definition

Consider a measure space. The characteristic function of any measurable set is measurable. A simple function is measurable if and only if each element of its simple partition is measurable.

The integral of a measurable non-negative simple function is the sum of the products of the measure of each piece with the value of the function on that piece. For example, the integral of a measurable characteristic function of a subset is the measure of that subset.

The integral operator is the real-valued function which associates each measurable non-negative simple function with its integral. The simple integral is non-negative, so the integral operator is a non-negative function.

Notation

Suppose $(X, \mathcal{A} , \mu )$ is a measure space and $f: X \to \R $ is a measurable simple function; i.e., there are $A_1, \dots , A_n \in \mathcal{A} $ and $a_1, \dots , a_n \in \R $ with:

\[ f = \sum_{i = 1}^{n} a_i \chi _{A_i}. \]

We denote the integral of $f$ with respect to measure $\mu $ by $\int f d\mu $, which we defined as

\[ \int f d\mu := \sum_{i = 1}^{n} a_i \mu (A_i). \]