Simple Functions, Nonnegative Simple Functions, and the Simple Integral

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space and $\mu$ a measure on $(X,\mathcal{F})$. For a map $s:X\to\mathbb{R}$, $s[X]$ is its image, a subset of $\mathbb{R}$, and for $B\subseteq\mathbb{R}$, $s^{-1}(B)$ is the preimage of $B$ under $s$.

(i) A simple function on $(X,\mathcal{F})$ is a map $s:X\to\mathbb{R}$ that is measurable as a map into $\mathbb{R}$ and whose image $s[X]$ is finite.

(ii) $\mathcal{S}(X,\mathcal{F})$ is the set of those $s\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Measurable Map, and the Set of Measurable Maps (ii)) with finite image $s[X]$, a predicative condition on $s$ by Induction over Finite Sets, and Predicative Formulas for the Segments [n] and for Finiteness (ii) and Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (i), so a set by Sets and Maps: Ordinary Notation (iii); as $\mathcal{B}(\mathbb{R})$ is a $\sigma$-algebra on $\mathbb{R}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), Borel Sigma-Algebra of a Metric Space (ii)), its elements are exactly the simple functions on $(X,\mathcal{F})$, by Measurable Map, and the Set of Measurable Maps (ii).

(iii) $\mathcal{S}^{+}(X,\mathcal{F})$ is the set of those $s\in\mathcal{S}(X,\mathcal{F})$ with $s(x)\ge0$ for every $x\in X$, the nonnegative simple functions on $(X,\mathcal{F})$.

(iv) Each $s\in\mathcal{S}^{+}(X,\mathcal{F})$ is also a map from $X$ to $[0,\infty]$, its values being nonnegative reals (The Extended Real Line, Its Order, and [0,∞] (ii)), with finite image $s[X]$, and $s^{-1}(\{c\})\in\mathcal{F}$ for every $c\in[0,\infty]$: for $c\in s[X]$ since $\{c\}\in\mathcal{B}(\mathbb{R})$ (Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (iii) on the metric space $(\mathbb{R},d_{\mathbb{R}})$ of Borel Sigma-Algebra of a Metric Space (ii)) and $s$ is measurable (Measurable Map, and the Set of Measurable Maps (iii), Measurable Map, and the Set of Measurable Maps (i)), and otherwise $s^{-1}(\{c\})=\emptyset\in\mathcal{F}$ (Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (ii), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vi)). The simple integral with respect to $\mu$ is the map $I_{\mu}:\mathcal{S}^{+}(X,\mathcal{F})\to[0,\infty]$ with

$$I_{\mu}(s)=\sum_{c\in s[X]}c\cdot\mu\bigl(s^{-1}(\{c\})\bigr)\qquad\text{for every }s\in\mathcal{S}^{+}(X,\mathcal{F}),$$

the sum over the finite set $s[X]$ formed with the sum $+$ of Addition and Multiplication on [0,∞] (i) and the product $\cdot$ of Addition and Multiplication on [0,∞] (ii); it exists and is unique by Operations over the Finite Images of Maps, and the Summands of the Simple Integral (i) and Operations over the Finite Images of Maps, and the Summands of the Simple Integral (ii), $+$ being associative and commutative with neutral element $0$ (Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i)). The value $I_{\mu}(s)$ is the simple integral of $s$ with respect to $\mu$.