Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$, $(Y,\mathcal{G})$ and $(Z,\mathcal{H})$ be measurable spaces. Maps that are $\mathcal{F}/\mathcal{G}$-measurable are as in Measurable Map, and the Set of Measurable Maps (i), and maps from $X$ that are measurable as maps into $\mathbb{R}$ or into $[0,\infty]$ are as in Measurable Map, and the Set of Measurable Maps (iii) and Measurable Map, and the Set of Measurable Maps (vi), with $\mathcal{B}([0,\infty])$ the Borel $\sigma$-algebra of $[0,\infty]$. On $[0,\infty]$ the relations $<$ and $>$ are those of The Extended Real Line, Its Order, and [0,∞] (iii).

(i) Let $\mathcal{C}$ be a subset of the power set $\mathcal{P}(Y)$ with $\mathcal{G}=\sigma(\mathcal{C})$. A map $f:X\to Y$ is $\mathcal{F}/\mathcal{G}$-measurable if and only if $f^{-1}(C)\in\mathcal{F}$ for every $C\in\mathcal{C}$.

(ii) If $f:X\to Y$ is $\mathcal{F}/\mathcal{G}$-measurable and $g:Y\to Z$ is $\mathcal{G}/\mathcal{H}$-measurable, then $g\circ f:X\to Z$ is $\mathcal{F}/\mathcal{H}$-measurable.

(iii) For every $y_{0}\in Y$, the map $X\to Y$ with value $y_{0}$ at every point is $\mathcal{F}/\mathcal{G}$-measurable, and the identity $\mathrm{id}_{X}$ is $\mathcal{F}/\mathcal{F}$-measurable.

(iv) Let $(M,d)$ and $(N,e)$ be metric spaces with Borel $\sigma$-algebras $\mathcal{B}(M,d)$ and $\mathcal{B}(N,e)$; by Borel Sigma-Algebra of a Metric Space (i) these are the $\sigma$-algebras generated by the open subsets of $M$ and of $N$, so they are $\sigma$-algebras on $M$ and on $N$, and $(M,\mathcal{B}(M,d))$ and $(N,\mathcal{B}(N,e))$ are measurable spaces, by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii). Let $h:M\to N$ be continuous. Then $h$ is $\mathcal{B}(M,d)/\mathcal{B}(N,e)$-measurable.

(v) For a map $f:X\to\mathbb{R}$ the following are equivalent: (a) $f$ is measurable as a map into $\mathbb{R}$; (b) $\{x\in X:f(x)>a\}\in\mathcal{F}$ for every $a\in\mathbb{R}$; (c) $\{x\in X:f(x)\ge a\}\in\mathcal{F}$ for every $a\in\mathbb{R}$; (d) $\{x\in X:f(x)<a\}\in\mathcal{F}$ for every $a\in\mathbb{R}$; (e) $\{x\in X:f(x)\le a\}\in\mathcal{F}$ for every $a\in\mathbb{R}$.

(vi) A map $f:X\to[0,\infty]$ is measurable as a map into $[0,\infty]$ if and only if $\{x\in X:f(x)>a\}\in\mathcal{F}$ for every $a\in\mathbb{R}$.

(vii) Let $f:X\to\mathbb{R}$ satisfy $f(x)\ge0$ for every $x\in X$; then $f(x)\in[0,\infty]$ for every $x\in X$ by The Extended Real Line, Its Order, and [0,∞] (ii), so $f$ is also a map into $[0,\infty]$. Then $f$ is measurable as a map into $\mathbb{R}$ ($\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable) if and only if $f$ is measurable as a map into $[0,\infty]$ ($\mathcal{F}/\mathcal{B}([0,\infty])$-measurable).

(viii) Let $A\in\mathcal{F}$ and let $\mathcal{F}_{A}=\{B\in\mathcal{F}:B\subseteq A\}$, a $\sigma$-algebra on $A$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (xi). If $f:X\to Y$ is $\mathcal{F}/\mathcal{G}$-measurable, then the restriction $f|_{A}:A\to Y$ is $\mathcal{F}_{A}/\mathcal{G}$-measurable.