Step Functions on a Partition of a Closed Interval as Simple Functions

In the settings of Measurable Functions and the Lebesgue Integral: Notation and The Real Line: Standing Notation and Background for Calculus on Intervals, assume the axiom of dependent choice. Let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval, and let $\lambda$ be Lebesgue measure on $\mathbb{R}$. Write $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ for the restriction of $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ to $[a,b]$ (a measure space, as $\lambda$ is a measure and $[a,b]\in\mathcal{B}(\mathbb{R})$ by Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv) and Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)); thus $\mathcal{B}_{[a,b]}=\{D\in\mathcal{B}(\mathbb{R}):D\subseteq[a,b]\}$ and $\lambda_{[a,b]}(D)=\lambda(D)$ for $D\in\mathcal{B}_{[a,b]}$. Let $P$ be a partition of $[a,b]$, and let $n\in\mathbb{N}$ and the increasing listing $a=x_{1}<x_{2}<\dots<x_{n+1}=b$ of $P$ be as in The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (i); $[n]=\{1,\dots,n\}$ is as in Intervals of Natural Numbers (ii). For $i\in[n]$, let $E_{i}$ be the open interval

$$E_{i}=(x_{i},x_{i+1}),$$

where $x_{i}$ and $x_{i+1}$ are defined by The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (ii).

(i) The cells cover the interval off the partition. For every $t\in[a,b]$ with $t\notin P$ there is $i\in[n]$ with $t\in E_{i}$.

(ii) Step functions exist. Let $c:[n]\to\mathbb{R}$, $i\mapsto c_{i}$, be a function and let $d\in\mathbb{R}$. There is a function $s:[a,b]\to\mathbb{R}$ with $s(t)=c_{i}$ for every $i\in[n]$ and every $t\in E_{i}$, and $s(t)=d$ for every $t\in P$.

(iii) The simple integral of a step function. Let $c:[n]\to\mathbb{R}$, $i\mapsto c_{i}$, be a function with $c_{i}\ge0$ for every $i\in[n]$, and let $s:[a,b]\to\mathbb{R}$ be a function with $s(t)\ge0$ for every $t\in[a,b]$ and $s(t)=c_{i}$ for every $i\in[n]$ and every $t\in E_{i}$; its values on $P$ are arbitrary. Then $s$ is a nonnegative simple function on $([a,b],\mathcal{B}_{[a,b]})$, and its simple integral is the real number

$$I_{\lambda_{[a,b]}}(s)=\sum_{i=1}^{n}c_{i}\,(x_{i+1}-x_{i}),$$

the finite sum in $\mathbb{R}$.