Simple functions

Definition: Simple function, canonical representation of simple functions

A real-valued function $\varphi :E\to \mathbb{R}$ is simple if it is measurable and takes a finite number of values. A simple function $\varphi$ that takes the distinct values $c_1,\dots ,c_n$ has the canonical representation

$$\varphi =\sum _{k=1}^n{c}_k\cdot {\chi }_{E_k}\text{ on E, where }{E}_k=\{x\in E|\varphi (x)=c_k\},$$

a linear combination of characteristic functions for each set $E_k$. In particular, this requires the $E_k$s to be disjoint and $c_k$s to be distinct.

Lemma: Approximation by simple functions

Let $f:E\to \mathbb{R}$ be measurable and bounded, meaning there exists some $M>0$ such that $|f(x)|\le M$ on $E$. Then for all $ϵ>0$, there exist simple functions ${\varphi }_{ϵ},{\psi }_{ϵ}:E\to \mathbb{R}$ such that

$${\varphi }_{ϵ}\le f\le {\psi }_{ϵ}\text{and}0\le {\psi }_{ϵ}-{\varphi }_{ϵ}<ϵ.$$

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Examples §

• Step functions, which take finite values, and each interval is measurable