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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 <math> 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 $f(X)=\{c_{1},\dots ,c_{n}\}$.

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