Nonnegative Measurable Maps and Null Sets: Constant Maps, the Pointwise-Sum Operation on 𝓜⁺, and Complements of Measure-Zero Sets

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation: for a $\sigma$-algebra $\mathcal{F}$ on a set $X$, $\mathcal{M}^{+}(X,\mathcal{F})$ is the set of measurable maps into $[0,\infty]$ of Nonnegative Measurable Maps and Their Lebesgue Integral (i); for maps $f,g:X\to[0,\infty]$, $f+g$ is their pointwise sum; and binary operations, their values $u\oplus v$ (Binary Operations on a Set (ii)), and associative and commutative binary operations and their neutral elements are as in Binary Operations on a Set and Associative and Commutative Binary Operations, and Neutral Elements.

(i) Let $\mathcal{F}$ be a $\sigma$-algebra on a set $X$, let $c\in[0,\infty]$, and let $g:X\to[0,\infty]$ be a map with $g(x)=c$ for every $x\in X$. Then $g\in\mathcal{M}^{+}(X,\mathcal{F})$.

(ii) Let $(X,\mathcal{F},\mu)$ be a measure space. There are a binary operation $\oplus$ on $\mathcal{M}^{+}(X,\mathcal{F})$ and a map $z:X\to[0,\infty]$ with $z(x)=0$ for every $x\in X$ such that $\oplus$ is associative and commutative, $z$ is a neutral element of $\oplus$, and

$$f\oplus g=f+g\qquad\text{for all }f,g\in\mathcal{M}^{+}(X,\mathcal{F}).$$

(iii) Let $(X,\mathcal{F},\mu)$ be a measure space, and let $Z\in\mathcal{F}$ with $\mu(Z)=0$. Then the difference $X\setminus Z$ holds $\mu$-almost everywhere.