Simple Functions

Why

We want to define area under a real function. We start with defining functions for which this notion is obvious.

Definition

A simple function (or step function) is a real-valued function whose range is a finite set. We can write simple function as the sum of the characteristic functions of the inverse image elements. We explain this more below.

We can partition the range of the function into a finite family of one-elements sets. Then the family whose members are the inverse images of these sets partitions the domain. We call this the simple partition of the function.

Notation

We denote the set of simple functions on $A$ by $\SimpleF(A)$. We denote subset of non-negative simple real functions with domain $A$ by by $\SimpleF_+(A)$.

Let $f \in \SimpleF(A)$. Let $\set{a_1, \dots , a_n} = f(A)$. Define $A_i = f^{-1}(\set{a_i})$. Then $f = \sum_{i = 1}^{n} a_i \chi _{A_i}$.