Proof of Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps

For a map $u:X\to\mathbb{R}$ and $a\in\mathbb{R}$, write $\{u>a\}=\{x\in X:u(x)>a\}$ and $\{u\ge a\}=\{x\in X:u(x)\ge a\}$, and likewise $\{u<a\}$ and $\{u\le a\}$. Let $\kappa:\mathbb{Q}\to\mathbb{R}$ be the canonical embedding of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (iv); for $a\in\mathbb{R}$ let $Q_{a}=\{w\in\mathbb{Q}:a<\kappa(w)\}$, a subset of $\mathbb{Q}$, and for $N\in\mathbb{N}$ let $T_{N}=\{l\in\mathbb{N}:l\ge N\}$. Here $>$, $\ge$, $<$ and $\le$ are those of $\mathbb{R}$; for real $u(x)$ and real $a$ they agree with those of $[-\infty,\infty]$ by Order and Suprema in the Extended Real Line (i), so these four sets are the superlevel and sublevel sets of Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (v), with $\mathbb{R}$ as a subset of $[-\infty,\infty]$. We use that for real numbers $b<b'$ there is $w\in\mathbb{Q}$ with $b<\kappa(w)<b'$, by Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema (iii), which applies to $\mathbb{R}$ with the canonical embedding $\kappa$ by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (vii). For $m\in\mathbb{N}$ and $x\in X$, both varying, $h_{m}(x)$ is the value at $(h_{m},x)$ of the evaluation map from $\mathbb{R}^{X}\times X$ to $\mathbb{R}$ (Projection, Evaluation and Currying Maps (ii)).

Countable index sets. $\mathbb{N}$ and $\mathbb{Q}$ are countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (i) and Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ix), so every $T_{N}$ and every $Q_{a}$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (iv). Each $T_{N}$ is nonempty: $N\in T_{N}$, since $N\le N$ for the order of $\mathbb{N}_{0}$, a well-order by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (v); and $\mathbb{N}$ is nonempty, as $1\in\mathbb{N}$. As each $T_{N}$ is a subset of $\mathbb{N}$, $N\mapsto T_{N}$ is a map $T$ from $\mathbb{N}$ to $\mathcal{P}(\mathbb{N})$ (Maps and Relations Given by Formulas (i)), and its value $T_{N}=T(N)$ is a term of type $\mathcal{P}(\mathbb{N})$ under the guard $N\in\mathbb{N}$ (Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them (iii)); it is the index set of the intersections over $T_{N}$ below, and $N$ is a term of type $\mathbb{N}$ under every guard that implies $N\in\mathbb{N}$ (Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them (i)). The sets $\mathbb{N}$ and $Q_{a}$, as parameters, are terms of type $\mathcal{P}(\mathbb{N})$ and $\mathcal{P}(\mathbb{Q})$ (Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them (ii)). For a single union or intersection of a map from one of these countable sets to $\mathcal{F}$ we use Unions and Intersections over a Countable Index Set in a σ-Algebra (i) and Unions and Intersections over a Countable Index Set in a σ-Algebra (ii), the latter for the nonempty index sets $\mathbb{N}$ and $T_{N}$. Nested unions and intersections are formed from the inside out, one index variable at a time: if $t$ is a term of type $\mathcal{F}$ under a guard expressing $\gamma\wedge i\in I$, where $I$ is one of these index sets, as a term of type $\mathcal{P}(\mathbb{N})$ or $\mathcal{P}(\mathbb{Q})$ under $\gamma$, with the countable set $\mathbb{N}$ or $\mathbb{Q}$, then $\bigcup_{i\in I}t$ is a term of type $\mathcal{F}$ under $\gamma$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (i), and, $I$ being $\mathbb{N}$ or $T_{N}$, so is $\bigcap_{i\in I}t$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (ii). At the outermost level, with no index variable left, the value of the term lies in $\mathcal{F}$. The innermost terms $\{h_{k}>\kappa(w)\}$ and $V_{jmn}$ below are of type $\mathcal{P}(X)$ by Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (v), $h_{k}$, $\kappa(w)$, $u_{mn}$ and $1/j$ being terms of types $\mathbb{R}^{X}$ and $\mathbb{R}$ (Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them (iii)), and their values lie in $\mathcal{F}$, so they are of type $\mathcal{F}$ by Terms and Typed Terms under a Guard (ii).

Suprema and infima. For every $m\in\mathbb{N}$ and $b\in\mathbb{R}$, $\{h_{m}>b\}$, $\{h_{m}\le b\}$ and $\{h_{m}\ge b\}$ belong to $\mathcal{F}$, by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v), (a) implying (b), (e) and (c), $h_{m}$ being measurable. Suppose first that every $A_{x}=\{h_{m}(x):m\in\mathbb{N}\}$ is bounded above; $A_{x}$ is nonempty, as $1\in\mathbb{N}$, so it has a supremum $s(x)$ in $\mathbb{R}$ by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (vii). Let $a\in\mathbb{R}$ and $x\in X$. If $s(x)\le a$, then $h_{m}(x)\le s(x)\le a$ for every $m$, as $s(x)$ is an upper bound of $A_{x}$. If $h_{m}(x)\le a$ for every $m$, then $a$ is an upper bound of $A_{x}$, so $s(x)\le a$ by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv). Hence

$$\{s\le a\}=\bigcap_{m\in\mathbb{N}}\{h_{m}\le a\},$$

the intersection of the map $m\mapsto\{h_{m}\le a\}$ from $\mathbb{N}$ to $\mathcal{F}$, its values lying in $\mathcal{F}$ (Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (vi), Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (v)), which belongs to $\mathcal{F}$ by Unions and Intersections over a Countable Index Set in a σ-Algebra (ii), $\mathbb{N}$ being countable and nonempty; so $s$ is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v), (e) implying (a).

Suppose next that every $A_{x}$ is bounded below, with infimum $\iota(x)$ in $\mathbb{R}$, which exists by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (viii). Let $a\in\mathbb{R}$ and $x\in X$. If $\iota(x)\ge a$, then $h_{m}(x)\ge\iota(x)\ge a$ for every $m$, as $\iota(x)$ is a lower bound of $A_{x}$; if $h_{m}(x)\ge a$ for every $m$, then $a$ is a lower bound of $A_{x}$, so $a\le\iota(x)$ by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv). Hence

$$\{\iota\ge a\}=\bigcap_{m\in\mathbb{N}}\{h_{m}\ge a\},$$

which belongs to $\mathcal{F}$ in the same way, and $\iota$ is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v), (c) implying (a). This proves (i).

Pointwise limits. Suppose that $(h_{m}(x))$ is convergent for every $x$, and let $L(x)=\lim_{m\to\infty}h_{m}(x)$, the real number to which it converges (The Limit of a Convergent Sequence (i)). Let $a\in\mathbb{R}$ and $x\in X$. We show that $L(x)>a$ if and only if there are $w\in Q_{a}$ and $N\in\mathbb{N}$ with $h_{k}(x)>\kappa(w)$ for every $k\in T_{N}$. If $L(x)>a$, take $w\in\mathbb{Q}$ with $a<\kappa(w)<L(x)$, so $w\in Q_{a}$, and let $\varepsilon=L(x)-\kappa(w)>0$; by Convergent Sequences of Real Numbers (i) there is $N\in\mathbb{N}$ with $|h_{k}(x)-L(x)|<\varepsilon$ for every $k\ge N$, and then $h_{k}(x)>L(x)-\varepsilon=\kappa(w)$. Conversely, if $w\in Q_{a}$, $N\in\mathbb{N}$ and $\kappa(w)\le h_{k}(x)$ for every $k\in T_{N}$, then $\kappa(w)\le L(x)$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x), so $a<\kappa(w)\le L(x)$. Hence

$$\{L>a\}=\bigcup_{w\in Q_{a}}\ \bigcup_{N\in\mathbb{N}}\ \bigcap_{k\in T_{N}}\{h_{k}>\kappa(w)\},$$

which belongs to $\mathcal{F}$ by the paragraph on countable index sets, the terms being formed from the inside out: under the guard $w\in Q_{a}\wedge N\in\mathbb{N}$, the intersection over $k\in T_{N}$ of the term $\{h_{k}>\kappa(w)\}$ of type $\mathcal{F}$ is a term of type $\mathcal{F}$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (ii), $T_{N}=T(N)$ being nonempty; under the guard $w\in Q_{a}$, its union over $N\in\mathbb{N}$ is a term of type $\mathcal{F}$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (i); and its union over $w\in Q_{a}$, with no variable left, is a term of type $\mathcal{F}$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (i), whose value $\{L>a\}$ therefore lies in $\mathcal{F}$. So $L$ is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v), (b) implying (a), which proves (ii).

The convergence set. Let $C$ be the set of $x\in X$ for which $(h_{m}(x))$ is convergent. By Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences (v), $x\in C$ if and only if $(h_{m}(x))$ is a Cauchy sequence. We show that this holds if and only if for every $j\in\mathbb{N}$ there is $N\in\mathbb{N}$ with $|h_{m}(x)-h_{n}(x)|<1/j$ for all $m,n\in T_{N}$. The forward direction is Cauchy Sequences of Real Numbers (i) with $\varepsilon=1/j$, which is positive since $0<j$ in $\mathbb{R}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi) and hence $0<1/j$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (viii), with $x=y=j$. Conversely, given a real $\varepsilon>0$, there is $j\in\mathbb{N}$ with $1/\varepsilon<j$ by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ix), so $1/j<\varepsilon$, and the $N$ belonging to this $j$ serves for $\varepsilon$. For $m,n\in\mathbb{N}$ the map $u_{mn}=|h_{m}-h_{n}|$, $x\mapsto|h_{m}(x)-h_{n}(x)|$, is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), so for $j\in\mathbb{N}$ the set $V_{jmn}=\{u_{mn}<1/j\}=\{x\in X:u_{mn}(x)<1/j\}$ belongs to $\mathcal{F}$ by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v), (a) implying (d). With $m$ and $n$ varying, $u_{mn}$ is the map $x\mapsto|h_{m}(x)-h_{n}(x)|$ given by an expression in the values $h_{m}(x)$ and $h_{n}(x)$ of the evaluation map (Projection, Evaluation and Currying Maps (ii)), so it is a term of type $\mathbb{R}^{X}$ (Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (vi)); and $1/j$, the value at $j$ of the map $j\mapsto1/j$ from $\mathbb{N}$ to $\mathbb{R}$ given by this expression ($j\neq0$, as $0<j$), is a term of type $\mathbb{R}$ under the variable $j\in\mathbb{N}$ (Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them (iii)). Hence

$$C=\bigcap_{j\in\mathbb{N}}\ \bigcup_{N\in\mathbb{N}}\ \bigcap_{m\in T_{N}}\ \bigcap_{n\in T_{N}}V_{jmn},$$

which belongs to $\mathcal{F}$ by the paragraph on countable index sets, the terms being formed from the inside out: under the guard $j\in\mathbb{N}\wedge N\in\mathbb{N}\wedge m\in T_{N}$, the intersection over $n\in T_{N}$ of the term $V_{jmn}$ of type $\mathcal{F}$ is a term of type $\mathcal{F}$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (ii); under the guard $j\in\mathbb{N}\wedge N\in\mathbb{N}$, so is its intersection over $m\in T_{N}$, by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (ii), $T_{N}=T(N)$ being nonempty in both steps; under the guard $j\in\mathbb{N}$, so is the union of the latter over $N\in\mathbb{N}$, by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (i); and its intersection over $j\in\mathbb{N}$, with no variable left, is a term of type $\mathcal{F}$ by Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms (ii), whose value $C$ therefore lies in $\mathcal{F}$. This proves (iii).