Properties of Simple Functions and of the Simple Integral

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space. Let $\mathcal{S}=\mathcal{S}(X,\mathcal{F})$ and $\mathcal{S}^{+}=\mathcal{S}^{+}(X,\mathcal{F})$ be the sets of simple functions and of nonnegative simple functions of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii) and Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii), and $I_{\mu}:\mathcal{S}^{+}\to[0,\infty]$ the simple integral with respect to $\mu$ of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv). For $A\subseteq X$, $\mathbf{1}_{A}$ is the indicator function of $A$. For $s,t\in\mathcal{S}$, the maps $s+t$, $st$ and $cs$, for $c\in\mathbb{R}$, are the pointwise sum, product and scalar multiple, and $\max(s,t)$ and $\min(s,t)$ are the maps $X\to\mathbb{R}$ with values $\max\{s(x),t(x)\}$ and $\min\{s(x),t(x)\}$ at $x\in X$, formed in $\mathbb{R}$ as in The Maximum and Minimum of Two Elements of a Total Order (i); $s\le t$ means $s(x)\le t(x)$ for every $x\in X$.

(i) For every $A\in\mathcal{F}$, $\mathbf{1}_{A}\in\mathcal{S}^{+}$ and $I_{\mu}(\mathbf{1}_{A})=\mu(A)$.

(ii) Let $s,t\in\mathcal{S}$ and $c\in\mathbb{R}$. Then $s+t$, $st$, $cs$, $\max(s,t)$ and $\min(s,t)$ belong to $\mathcal{S}$; if moreover $s,t\in\mathcal{S}^{+}$ and $c\ge0$, they belong to $\mathcal{S}^{+}$.

(iii) Let $I$ be a finite set, let $i\mapsto c_{i}$ be a map from $I$ to $\mathbb{R}$ with $c_{i}\ge0$ for every $i\in I$, and let $i\mapsto A_{i}$ be a map from $I$ to $\mathcal{F}$ with $A_{i}\cap A_{j}=\emptyset$ for all $i,j\in I$ with $i\neq j$. Let $s:X\to\mathbb{R}$ satisfy $s(x)=\sum_{i\in I}c_{i}\mathbf{1}_{A_{i}}(x)$ for every $x\in X$, the sum over $I$, formed in $\mathbb{R}$, of the map $i\mapsto c_{i}\mathbf{1}_{A_{i}}(x)$ (which exists, 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)). Then $s\in\mathcal{S}^{+}$ and $I_{\mu}(s)=\sum_{i\in I}c_{i}\cdot\mu(A_{i})$, the sum and the products in $[0,\infty]$.

(iv) For all $s,t\in\mathcal{S}^{+}$, $I_{\mu}(s+t)=I_{\mu}(s)+I_{\mu}(t)$.

(v) For every $s\in\mathcal{S}^{+}$ and every $c\in\mathbb{R}$ with $c\ge0$, $I_{\mu}(cs)=c\cdot I_{\mu}(s)$.

(vi) For all $s,t\in\mathcal{S}^{+}$ with $s\le t$, $I_{\mu}(s)\le I_{\mu}(t)$.

(vii) Let $s\in\mathcal{S}^{+}$; for $A\in\mathcal{F}$, $s\mathbf{1}_{A}\in\mathcal{S}^{+}$ by (i) and (ii). The map $\nu:\mathcal{F}\to[0,\infty]$, $A\mapsto I_{\mu}(s\mathbf{1}_{A})$, is a measure on $(X,\mathcal{F})$.