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 $\mathbb {R}$. 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 sets $E_{i}=f^{-1}(\{c_{i}\})\in {\mathcal {M}}$ that partition $X$, where $f(X)=\{c_{1},\dots ,c_{n}\}$.

Integration of Simple Functions

These functions earn their name from the simplicity in which their integrals are defined. Let $L^{+}$ be the space of all measurable functions from $X$ to $[0,+\infty ].$ Then

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

where by convention, we let $0\cdot \infty =0$. Note that $d\mu (x)$ is equivalent to $\mu (dx)$ and that some arguments may be omitted when there is no confusion.

Furthermore, for any $A\in {\mathcal {M}}$, we define

$\int _{A}f(x)d\mu (x)=\int _{X}f(x)1_{A}d\mu (x)=\sum _{i=1}^{n}c_{i}\mu (E_{i}\cap A)$

${\displaystyle }$