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

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space. Sums and products in $[0,\infty]$ are those of Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii), and sums over a finite set of elements of $[0,\infty]$ are as in Sums and Products over a Finite Set and over an Interval (v), $([0,\infty],+,0)$ being a commutative monoid by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i). Let $c\in\mathbb{R}$ with $c\ge0$, and let $c\mu:\mathcal{F}\to[0,\infty]$ be the function

$$(c\mu)(A)=c\cdot\mu(A)\qquad(A\in\mathcal{F}).$$

(i) Finite sums. Let $I$ be a finite set and $\Phi:I\times X\to[0,\infty]$ a function such that, for every $i\in I$, the function $f_{i}=\Phi(i,\cdot)$, $x\mapsto\Phi(i,x)$, lies in $\mathcal{M}^{+}(X,\mathcal{F})$. Let $\sum_{i\in I}f_{i}:X\to[0,\infty]$ be the function with

$$\Bigl(\sum_{i\in I}f_{i}\Bigr)(x)=\sum_{i\in I}\Phi(i,x)\qquad(x\in X).$$

Then $\sum_{i\in I}f_{i}\in\mathcal{M}^{+}(X,\mathcal{F})$ and

$$\int_{X}\sum_{i\in I}f_{i}\,d\mu=\sum_{i\in I}\int_{X}f_{i}\,d\mu .$$

(ii) Constant multiples of a measure. $c\mu$ is a measure on $(X,\mathcal{F})$, and $c\mu=g\mu$, the measure with density $g$ with respect to $\mu$, for the constant function $g:X\to[0,\infty]$ with value $c$.

(iii) Integrals of nonnegative functions against $c\mu$. For every $f\in\mathcal{M}^{+}(X,\mathcal{F})$,

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

(iv) Integrability with respect to $c\mu$. Let $c>0$ and let $f:X\to\mathbb{R}$ be measurable. Then $f\in\mathcal{L}^{1}(c\mu)$ if and only if $f\in\mathcal{L}^{1}(\mu)$, and in that case

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