Proof of Integrals of Finite Sums of Nonnegative Functions, and Constant Multiples of a Measure

Proof of (i). For a subset $C$ of $I$ let $\sum_{i\in C}f_{i}$ be the function $x\mapsto\sum_{i\in C}\Phi(i,x)$. We show by induction over finite sets (Induction over Finite Sets, and Predicative Formulas for the Segments [n] and for Finiteness (iii), the property being predicative by Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices (iv)) that every finite $C$ with $C\subseteq I$ satisfies

$$\sum_{i\in C}f_{i}\in\mathcal{M}^{+}(X,\mathcal{F}),\qquad\int_{X}\sum_{i\in C}f_{i}\,d\mu=\sum_{i\in C}\int_{X}f_{i}\,d\mu ;$$

the case $C=I$ is the claim. For $C=\emptyset$ both sums are empty and equal to $0$ (Sums and Products over a Finite Set and over an Interval (ii)), and the zero function lies in $\mathcal{M}^{+}(X,\mathcal{F})$ with integral $0$ (Basic Properties of the Integral of Nonnegative Measurable Maps (v)). Let $C$ satisfy the claim and let $j\notin C$ with $C\cup\{j\}\subseteq I$. Splitting off the term at $j$ (Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (ii), Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (i)),

$$\sum_{i\in C\cup\{j\}}f_{i}=\sum_{i\in C}f_{i}+f_{j},$$

and likewise for the sum of the integrals. By additivity of the integral on $\mathcal{M}^{+}$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (i)), $\sum_{i\in C\cup\{j\}}f_{i}\in\mathcal{M}^{+}(X,\mathcal{F})$ and

$$\int_{X}\sum_{i\in C\cup\{j\}}f_{i}\,d\mu=\int_{X}\sum_{i\in C}f_{i}\,d\mu+\int_{X}f_{j}\,d\mu=\sum_{i\in C}\int_{X}f_{i}\,d\mu+\int_{X}f_{j}\,d\mu=\sum_{i\in C\cup\{j\}}\int_{X}f_{i}\,d\mu .$$

Proof of (ii). The constant function $g$ lies in $\mathcal{M}^{+}(X,\mathcal{F})$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)). For $A\in\mathcal{F}$ we have $(g\mathbf{1}_{A})(x)=c\cdot\mathbf{1}_{A}(x)$ for every $x\in X$, so (Integrals over Measurable Subsets (i), Basic Properties of the Integral of Nonnegative Measurable Maps (iv), Basic Properties of the Integral of Nonnegative Measurable Maps (ii))

$$(g\mu)(A)=\int_{A}g\,d\mu=\int_{X}g\mathbf{1}_{A}\,d\mu=c\int_{X}\mathbf{1}_{A}\,d\mu=c\cdot\mu(A)=(c\mu)(A).$$

Thus $c\mu=g\mu$, which is a measure on $(X,\mathcal{F})$ by Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (v).

Proof of (iii). Let $g$ be the constant function of (ii), so that $c\mu=g\mu$. For $f\in\mathcal{M}^{+}(X,\mathcal{F})$ the product $fg$ has the values $f(x)\,c=c\,f(x)$ (Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (ii)), so Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vi) and Basic Properties of the Integral of Nonnegative Measurable Maps (iv) give

$$\int_{X}f\,d(c\mu)=\int_{X}fg\,d\mu=c\int_{X}f\,d\mu .$$

Proof of (iv). With $g$ as before, $c\mu=g\mu$ by (ii). As $g$ takes the real value $c$ everywhere, Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (vii) applies: $f\in\mathcal{L}^{1}(c\mu)$ if and only if the real product $fg=cf$ lies in $\mathcal{L}^{1}(\mu)$, and then $\int_{X}f\,d(c\mu)=\int_{X}cf\,d\mu$. By linearity (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii)), $cf\in\mathcal{L}^{1}(\mu)$ with $\int_{X}cf\,d\mu=c\int_{X}f\,d\mu$ whenever $f\in\mathcal{L}^{1}(\mu)$, and conversely $f=c^{-1}(cf)\in\mathcal{L}^{1}(\mu)$ whenever $cf\in\mathcal{L}^{1}(\mu)$.