Measurable Map, and the Set of Measurable Maps

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be measurable spaces, and let $n,p,q\in\mathbb{N}$.

(i) A map $f:X\to Y$ is $\mathcal{F}/\mathcal{G}$-measurable if $f^{-1}(B)\in\mathcal{F}$ for every $B\in\mathcal{G}$, where $f^{-1}(B)$ is the preimage $f^{-1}[B]$ of $B$ under $f$.

(ii) $\mathrm{Meas}(X,\mathcal{F};Y,\mathcal{G})$ is the set of those elements $f$ of the set $Y^{X}$ of maps from $X$ to $Y$ that are $\mathcal{F}/\mathcal{G}$-measurable; $Y^{X}$ is a set by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii). By The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i), the elements of $\mathrm{Meas}(X,\mathcal{F};Y,\mathcal{G})$ are exactly the $\mathcal{F}/\mathcal{G}$-measurable maps $f:X\to Y$.

Let $\mathcal{B}(\mathbb{R})$, $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R}^{p\times q})$ be as in Borel Sigma-Algebra of a Metric Space (ii), Borel Sigma-Algebra of a Metric Space (iii) and Borel Sigma-Algebra of a Metric Space (iv), and $\mathcal{B}([0,\infty])$ as in The Borel σ-Algebra of [0,∞] (ii). Each is a generated $\sigma$-algebra, hence a $\sigma$-algebra on $\mathbb{R}$, $\mathbb{R}^{n}$, $\mathbb{R}^{p\times q}$ and $[0,\infty]$ respectively, by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii); so $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, $(\mathbb{R}^{p\times q},\mathcal{B}(\mathbb{R}^{p\times q}))$ and $([0,\infty],\mathcal{B}([0,\infty]))$ are measurable spaces.

In the four clauses below, the word measurable is relative to the target named: a map is measurable as a map into $\mathbb{R}$, $\mathbb{R}^{n}$, $\mathbb{R}^{p\times q}$ or $[0,\infty]$. A map $f:X\to\mathbb{R}$ with $0\le f(x)$ for every $x\in X$ has all its values in $[0,\infty]$, by The Extended Real Line, Its Order, and [0,∞] (ii), so it is also a map into $[0,\infty]$; for such a map, "measurable as a map into $\mathbb{R}$" and "measurable as a map into $[0,\infty]$" are the two readings, and the target is always named.

(iii) A map $f:X\to\mathbb{R}$ is measurable as a map into $\mathbb{R}$ if it is $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable.

(iv) A map $f:X\to\mathbb{R}^{n}$ is measurable as a map into $\mathbb{R}^{n}$ if it is $\mathcal{F}/\mathcal{B}(\mathbb{R}^{n})$-measurable.

(v) A map $f:X\to\mathbb{R}^{p\times q}$ is measurable as a map into $\mathbb{R}^{p\times q}$ if it is $\mathcal{F}/\mathcal{B}(\mathbb{R}^{p\times q})$-measurable.

(vi) A map $f:X\to[0,\infty]$ is measurable as a map into $[0,\infty]$ if it is $\mathcal{F}/\mathcal{B}([0,\infty])$-measurable.