Basic Properties of the Integral of Nonnegative Measurable Maps

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 $\mathcal{M}^{+}=\mathcal{M}^{+}(X,\mathcal{F})$ be the set of measurable maps into $[0,\infty]$ of Nonnegative Measurable Maps and Their Lebesgue Integral (i), and $\int_{X}f\,d\mu$, for $f\in\mathcal{M}^{+}$, the integral of Nonnegative Measurable Maps and Their Lebesgue Integral (ii). Let $\mathcal{S}^{+}=\mathcal{S}^{+}(X,\mathcal{F})$ be the set of nonnegative simple functions of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii), and $I_{\mu}$ the simple integral of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv); an element of $\mathcal{S}^{+}$ has values in $[0,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), so it is also a map $X\to[0,\infty]$. For $A\subseteq X$, $\mathbf{1}_{A}$ is the indicator function of $A$.

(i) $\mathcal{S}^{+}\subseteq\mathcal{M}^{+}$, and $\int_{X}s\,d\mu=I_{\mu}(s)$ for every $s\in\mathcal{S}^{+}$.

(ii) For every $A\in\mathcal{F}$, $\mathbf{1}_{A}\in\mathcal{M}^{+}$ and $\int_{X}\mathbf{1}_{A}\,d\mu=\mu(A)$.

(iii) For all $f,g\in\mathcal{M}^{+}$ with $f(x)\le g(x)$ for every $x\in X$, $\int_{X}f\,d\mu\le\int_{X}g\,d\mu$.

(iv) Let $c\in\mathbb{R}$ with $c\ge0$, so that $c\in[0,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), and let $f\in\mathcal{M}^{+}$. Every map $h:X\to[0,\infty]$ with $h(x)=c\cdot f(x)$ for every $x\in X$, the product of Addition and Multiplication on [0,∞] (ii), belongs to $\mathcal{M}^{+}$ and satisfies $\int_{X}h\,d\mu=c\cdot\int_{X}f\,d\mu$.

(v) Every map $z:X\to[0,\infty]$ with $z(x)=0$ for every $x\in X$ belongs to $\mathcal{M}^{+}$ and satisfies $\int_{X}z\,d\mu=0$.