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

Conventions. Write $\mathcal{B}'=\mathcal{B}_{[a,b]}$ and $\lambda'=\lambda_{[a,b]}$, and for $D\subseteq[a,b]$ let $\mathbf{1}_{D}:[a,b]\to\mathbb{R}$ be the indicator function of $D$. By Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i) and Integrals of Functions over a Measurable Subset (i), $\lambda'$ is a measure on $([a,b],\mathcal{B}')$ with $\lambda'(D)=\lambda(D)$ for $D\in\mathcal{B}'$. Sums over a finite set of real numbers are formed with the sum of $\mathbb{R}$, associative and commutative with neutral element $0$ (Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (iii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v)).

Step 1: the cells. Let $i\in[n]$. Then $x_{i},x_{i+1}\in P\subseteq[a,b]$ and $x_{i}<x_{i+1}$ (The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (ii), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (iii)), so $E_{i}\subseteq[a,b]$; as $E_{i}\in\mathcal{B}(\mathbb{R})$, we get $E_{i}\in\mathcal{B}'$ and

$$\lambda'(E_{i})=\lambda\bigl((x_{i},x_{i+1})\bigr)=x_{i+1}-x_{i}$$

(Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)); and $i\mapsto E_{i}$ is a function from $[n]$ to $\mathcal{B}'$ (Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps (iii)). Two further facts hold. (a) No point of $P$ lies in $E_{i}$. By The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (i), each $p\in P$ is $x_{k}$ for some $k\in[n+1]$. If $k\le i$, then $x_{k}\le x_{i}$; otherwise $i<k$, so $i+1\le k$ and $x_{i+1}\le x_{k}$ (Arithmetic and Order of the Natural Numbers (vii), Arithmetic and Order of the Natural Numbers (xi), Arithmetic and Order of the Natural Numbers (vi)). In both cases $x_{k}\notin(x_{i},x_{i+1})$. (b) The cells are pairwise disjoint. For $i<j$ in $[n]$, $i+1\le j$, so $x_{i+1}\le x_{j}$, and every $t\in E_{i}$ has $t<x_{i+1}\le x_{j}$, so $t\notin E_{j}$ (same references).

Proof of (i). Let $t\in[a,b]$ with $t\notin P$. By The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (iv) there is $i\in[n]$ with $x_{i}\le t\le x_{i+1}$. As $x_{i},x_{i+1}\in P$ and $t\notin P$, $t\ne x_{i}$ and $t\neq x_{i+1}$, so $x_{i}<t<x_{i+1}$, that is, $t\in E_{i}$.

Proof of (ii). The set $[n]$ is finite (Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii)). Let $s:[a,b]\to\mathbb{R}$ be the function with

$$s(t)=d\,\mathbf{1}_{P}(t)+\sum_{i\in[n]}c_{i}\mathbf{1}_{E_{i}}(t)\qquad(t\in[a,b])$$

(it exists by Maps and Relations Given by Formulas (i), the summands $i\mapsto c_{i}\mathbf{1}_{E_{i}}(t)$ forming a function on $[n]$ by Maps Used to Build the Integral: Fibres of a Map, Pointwise Combinations with Finite Image, Tail Extrema of a Sequence of Maps, the Integer Part, and Indicators (v)). Let $j\in[n]$ and $t\in E_{j}$. Then $\mathbf{1}_{P}(t)=0$ by (a), and $\mathbf{1}_{E_{i}}(t)=0$ for $i\neq j$ by (b), so $s(t)=c_{j}$ (Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (vii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v)). Let $t\in P$. Then $\mathbf{1}_{P}(t)=1$, and $\mathbf{1}_{E_{i}}(t)=0$ for every $i\in[n]$ by (a), so $s(t)=d$ (Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii)).

Proof of (iii). Two simple functions. The set $P$ is finite (Partitions of a Closed Interval, the Set of Partitions, and Refinements (i)). For $p\in P$, $\{p\}=[p,p]\in\mathcal{B}(\mathbb{R})$ and $\{p\}\subseteq[a,b]$, so $\{p\}\in\mathcal{B}'$ and $\lambda'(\{p\})=p-p=0$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)); $p\mapsto\{p\}$ is a function from $P$ to $\mathcal{B}'$ (Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps (iii)), and distinct points have disjoint singletons. Let $u,v:[a,b]\to\mathbb{R}$ be the functions with

$$u(t)=\sum_{i\in[n]}c_{i}\mathbf{1}_{E_{i}}(t),\qquad v(t)=\sum_{p\in P}s(p)\,\mathbf{1}_{\{p\}}(t)\qquad(t\in[a,b])$$

(as in the proof of (ii)). As $c_{i}\ge0$, $s(p)\ge0$ and by (b), Properties of Simple Functions and of the Simple Integral (iii) gives $u,v\in\mathcal{S}^{+}([a,b],\mathcal{B}')$ with

$$I_{\lambda'}(u)=\sum_{i\in[n]}c_{i}\cdot(x_{i+1}-x_{i}),\qquad I_{\lambda'}(v)=\sum_{p\in P}s(p)\cdot0=0,$$

the sums and products in $[0,\infty]$ (Addition and Multiplication on [0,∞] (ii), Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii)).

The decomposition $s=u+v$. Let $t\in[a,b]$. If $t\in E_{j}$ for some $j\in[n]$, then $u(t)=c_{j}$ as in the proof of (ii), and $v(t)=0$, as $t\notin P$ by (a), so $\mathbf{1}_{\{p\}}(t)=0$ for every $p\in P$ (Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii)); so $u(t)+v(t)=c_{j}=s(t)$. If $t\in P$, then $u(t)=0$ by (a) and $v(t)=s(t)$ (Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii), Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iii)), so $u(t)+v(t)=s(t)$. By (i) these cases cover $[a,b]$, so $s$ is the pointwise sum $u+v$ (Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v)).

Conclusion. By Properties of Simple Functions and of the Simple Integral (ii) and Properties of Simple Functions and of the Simple Integral (iv), $s\in\mathcal{S}^{+}([a,b],\mathcal{B}')$ and

$$I_{\lambda'}(s)=I_{\lambda'}(u)+I_{\lambda'}(v)=I_{\lambda'}(u)$$

(Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i)). Each $c_{i}\cdot(x_{i+1}-x_{i})$ is a product of nonnegative reals, hence the real product $c_{i}(x_{i+1}-x_{i})\ge0$ (Addition and Multiplication on [0,∞] (ii), Rules of Arithmetic and Order in an Ordered Field (vi)), and a sum over $[n]$ of nonnegative reals formed in $[0,\infty]$ is their real sum, which is the finite sum $\sum_{i=1}^{n}$ (A Finite Sum of Nonnegative Real Numbers in [0,∞] Is Their Real Sum (i), Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v)). Hence

$$I_{\lambda'}(s)=\sum_{i=1}^{n}c_{i}\,(x_{i+1}-x_{i})\in\mathbb{R}.$$