Lebesgue Measurable Step Functions

Recall from the Lebesgue Measurable Simple Functions page that a Lebesgue measurable function $f$ is a simple function if the range of $f$ is finite.

Furthermore, if $E \subseteq F$ then the characteristic function of $E$ in $F$ is the function $\chi_E : F \to \{ 0, 1 \}$ defined for all $x \in F$ by:

(1)

\begin{align} \quad \chi_E (x) = \left\{\begin{matrix} 0 & \mathrm{if} \: x \in F \setminus E \\ 1 & \mathrm{if} \: x \in E \end{matrix}\right. \end{align}

We now define another important type of function known as a step function on an interval $[a, b]$.

Definition: Let $I = [a, b]$ be a closed and bounded interval. A function $f$ defined on $I$ is a Step Function if there exists a partition of $I$, say $P = \{ a_0, a_1, ..., a_n \} \in \wp [a, b]$ with $a = a_0 < a_1 < ... < a_n = b$ such that $f$ is constant on each subinterval $(a_k, a_{k+1})$ for $k = 0, 1, ..., n - 1$.

The notation "$\wp [a, b]$" denotes the set of all partitions on the interval $[a, b]$.

An example of a graph of a simple function on a closed interval $[a, b]$ is given below.