Maps Used to Build the Integral: Fibres of a Map, Pointwise Combinations with Finite Image, Tail Extrema of a Sequence of Maps, the Integer Part, and Indicators

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, maps and their values are as in Sets and Maps: Ordinary Notation (v), and for sets $A$ and $B$, $B^{A}$ is the set of maps from $A$ to $B$ of The Class of All Functions from One Set to Another (i) and $\mathcal{P}(A)$ the power set of $A$. For a map $f:A\to B$ and a subset $S$ of $A$, $f[S]$ is the image of $S$ under $f$, and for a set $V$, $f^{-1}(V)$ is the preimage of $V$ under $f$, as in Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (i). Finite sets are as in Finite Sets (i).

(i) Let $S$ and $T$ be sets and $u:S\to T$ a map. There is exactly one map $\varphi:T\to\mathcal{P}(S)$ with $\varphi(c)=u^{-1}(\{c\})$ for every $c\in T$, where $\{c\}$ is the singleton of $c$.

(ii) Let $X$ be a set, let $s,t:X\to\mathbb{R}$ be maps such that $s[X]$ and $t[X]$ are finite, and let $F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be a map. Let $h:X\to\mathbb{R}$ be the map with $h(x)=F(s(x),t(x))$ for every $x\in X$, which exists and is unique by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii). Then $h[X]\subseteq F[s[X]\times t[X]]$, and $h[X]$ and $F[s[X]\times t[X]]$ are finite.

(iii) Let $X$ be a set, let $[-\infty,\infty]$ be the extended real line, and let $(f_{m})$ be a sequence in $[-\infty,\infty]^{X}$. For $x\in X$, $(f_{m}(x))_{m}$ is the sequence $m\mapsto f_{m}(x)$ in $[-\infty,\infty]$, and its tail infima $\inf_{m\ge k}f_{m}(x)$ and tail suprema $\sup_{m\ge k}f_{m}(x)$, for $k\in\mathbb{N}$, its limit inferior and its limit superior are as in Tail Extrema, Limit Inferior and Limit Superior of a Sequence in [−∞,∞] (i), Tail Extrema, Limit Inferior and Limit Superior of a Sequence in [−∞,∞] (ii) and Tail Extrema, Limit Inferior and Limit Superior of a Sequence in [−∞,∞] (iii). There is exactly one sequence $(g_{k})$ in $[-\infty,\infty]^{X}$ with $g_{k}(x)=\inf_{m\ge k}f_{m}(x)$ for all $k\in\mathbb{N}$ and $x\in X$, and exactly one sequence $(h_{k})$ in $[-\infty,\infty]^{X}$ with $h_{k}(x)=\sup_{m\ge k}f_{m}(x)$ for all $k\in\mathbb{N}$ and $x\in X$. There is exactly one map $L:X\to[-\infty,\infty]$ with $L(x)=\sup\{g_{k}(x):k\in\mathbb{N}\}$ for every $x\in X$, and exactly one map $U:X\to[-\infty,\infty]$ with $U(x)=\inf\{h_{k}(x):k\in\mathbb{N}\}$ for every $x\in X$; for $x\in X$, $L(x)=\liminf_{m\to\infty}f_{m}(x)$ and $U(x)=\limsup_{m\to\infty}f_{m}(x)$.

(iv) For $z\in\mathbb{R}$, $\lfloor z\rfloor$ is the integer part of $z$, an integer and so a real number, as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v) and The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (vi). There is exactly one map $\lfloor\cdot\rfloor:\mathbb{R}\to\mathbb{R}$ with value $\lfloor z\rfloor$ at every $z\in\mathbb{R}$; and for every set $X$ and every map $y:X\to\mathbb{R}$ there is exactly one map $X\to\mathbb{R}$ with value $\lfloor y(x)\rfloor$ at every $x\in X$.

(v) Let $X$ be a set and, for $A\in\mathcal{P}(X)$, let $\mathbf{1}_{A}:X\to\mathbb{R}$ be the indicator function of $A$. There is exactly one map $I:\mathcal{P}(X)\to\mathbb{R}^{X}$ with $I(A)=\mathbf{1}_{A}$ for every $A\in\mathcal{P}(X)$.