Simple Function

The simplest functions you will ever integrate, hence the name.

Definition

Let $(X,{\mathcal {M}},\mu )$ be a measure space. A measurable function $f:X\rightarrow \mathbb {R}$ is a simple function if $f(X)$ is a finite subset of Failed to parse (syntax error): {\displaystyle \mathbb{R} <ref name="Craig">Craig, Katy. ''MATH 201A Lecture 11''. UC Santa Barbara, Fall 2020.</ref>} . The standard representation for a simple function is given by

$f(x)=\sum _{i=1}^{n}c_{i}1_{E_{i}}(x)$,

where $1_{E_{i}}(x)$ is the indicator function on the disjoint set $E_{i}=f^{-1}(\{c_{i}\})$ where $f(X)=\{c_{1},\dots ,c_{n}\}$.