Weak Convergence Tightness and Representation — Examples

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): The Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, is the following statement.

For every family $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets indexed by $\mathbb{N}$ there is a function $f$ with domain $\mathbb{N}$ such that $f(n)\in X_n$ for every $n\in \mathbb{N}$.

Equivalently, in the vocabulary of def-choice-function: every at most countable family of nonempty sets (def-countable) has a choice function.

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: Let $n\ge 1$, assume the Axiom of Countable Choice (def-countable-choice), and let $a_i\le b_i$ be reals for $i<n$. Write

$R^{\circ }:=\{x\in {\mathbb{R}}^n:a_i<x_i<b_i\text{ for every }i<n\},\bar{R}:=[a,b]=\{x\in {\mathbb{R}}^n:a_i\le x_i\le b_i\text{ for every }i<n\}$

(def-multidimensional-rectangle-and-volume). Then $R^{\circ }$ is open and $\bar{R}$ is closed, so both are Borel and Lebesgue measurable, and every set $R$ with $R^{\circ }\subseteq R\subseteq \bar{R}$ is Lebesgue measurable with

${\lambda }_n(R)={\prod }_{i<n}(b_i-a_i)\mathrm{.}$

In particular this covers the four one-dimensional face conventions in each coordinate — the open box, the closed box $[a,b]$, the half-open box $B(a,b)={\prod }_{i<n}(a_i,b_i]$ of def-half-open-box, and every mixture of them, in any combination of coordinates — and it gives measure $0$ to all of them whenever $a_i=b_i$ for some $i<n$. For a half-open box with infinite parameters the value is already ${\lambda }_n(B)=\mathrm{vol}(B)$ (thm-lebesgue-measure-is-a-complete-measure).

The nonnegative integral agrees with the simple integral on simple functions: If $s$ is a nonnegative simple measurable function, then its nonnegative Lebesgue integral equals its simple integral: $\int sd\mu ={\int }_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}sd\mu \mathrm{.}$

Heine-Cantor in $\mathbb{R}$: a continuous real function on a compact subset of $\mathbb{R}$ is uniformly continuous, proved $\mathbb{R}$-natively from sequential compactness: Let $K\subseteq \mathbb{R}$ be compact (def-open-cover-r) and let $f:K\to \mathbb{R}$ be continuous on $K$ (def-continuity-real). Then $f$ is uniformly continuous on $K$ (def-uniform-continuity-real).

This theorem is stated twice in this library, on purpose. Its metric-space twin is thm-heine-cantor-metric, proved there from the cover machinery of metric spaces; the proof below is $\mathbb{R}$-native and runs through thm-compact-iff-sequentially-compact-r, which is order-based. That the two statements are the same statement in two vocabularies is lem-real-and-metric-notions-agree, clauses 1, 2 and 5, immediately above.

The choice cost, named. The proof invokes the axiom of countable choice (def-countable-choice) exactly once, at step 3.1, to select one bad pair of points from each of countably many nonempty sets. The backward implication of thm-compact-iff-sequentially-compact-r also spends countable choice, and that item names its own uses; the forward implication used here, from compact to sequentially compact, does not. No claim is made that the axiom is necessary for either.

The Lebesgue integral is linear on $L^1(\mu )$: The class $L^1(\mu )$ is a complex vector space, and the Lebesgue integral is complex-linear on it: $\int (\alpha f+\beta g)d\mu =\alpha \int fd\mu +\beta \int gd\mu (\alpha ,\beta \in \mathbb{C},f,g\in L^1(\mu ))\mathrm{.}$

The modulus of an integral is bounded by the integral of the modulus: If $f\in L^1(\mu )$, then $\mid \int fd\mu \mid \le \int \mathrm{\mid }f\mathrm{\mid }d\mu \mathrm{.}$

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it: Let $n\in \mathbb{N}$ with $n\ge 1$. A von Neumann natural is the set of its predecessors, $n=\{0,1,\dots ,n-1\}$ (def-natural-numbers), so it can be used directly as an index set. Define

${\mathbb{R}}^n:=\{x:x\text{ is a function }n\to \mathbb{R}\},$

and write $x_k$ for $x(k)$, $k<n$. Two elements of ${\mathbb{R}}^n$ are equal exactly when they agree at every $k<n$, functions being equal when they have the same values. For $x,y\in {\mathbb{R}}^n$ put

$d_1(x,y):={\sum }_{k<n}\mathrm{\mid }{x}_k-y_k\mathrm{\mid },d_2(x,y):=\sqrt{{\sum }_{k<n}(x_k-y_k{)}^2},d_{\mathrm{\infty }}(x,y):=\max \{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}\mathrm{.}$

All three are well defined: the finite sums are those of def-finite-sum; the sum of squares is nonnegative (lem-finite-sum-laws, lem-of-square-positive) so it has a unique nonnegative square root (thm-of-square-roots); and $\{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}$ is a nonempty finite subset of $\mathbb{R}$, because $n\ge 1$, so it has a maximum (lem-finite-set-has-max, def-max-min).

Then $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ are metrics on ${\mathbb{R}}^n$ (def-metric-space).

Why $n\ge 1$. For $n=0$ the set ${\mathbb{R}}^0$ has exactly one element, the empty function, and $d_1$ and $d_2$ are the empty sum $0$ and its root; but $d_{\mathrm{\infty }}$ would be the maximum of the empty set, which does not exist. The hypothesis $n\ge 1$ is therefore not decoration, and it is carried by every statement about $d_{\mathrm{\infty }}$ in this library.

Markov's inequality for random variables: If $X:\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ is a nonnegative random variable on a probability space and $a>0$, then $\mathbb{P}(X\ge a)\le \frac{\mathbb{E}[X]}{a}\mathrm{.}$

The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents: For $a,b>0$ and $r,s\in \mathbb{R}$, $a^{r+s}=a^r{a}^s,(ab{)}^r=a^r{b}^r,(a\mathrm{/}b{)}^r=a^r\mathrm{/}{b}^r,(a^r{)}^s=a^{rs}\mathrm{.}$

Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: Let $n\in \mathbb{N}$ with $n\ge 1$, let ${\mathbb{R}}^n$ be the set of functions $n\to \mathbb{R}$ and let $d_2$ be the Euclidean metric on it (lem-metrics-on-rn). Then:

1. Closed boxes are compact. For reals $a_k\le b_k$ $(k<n)$ the box $Q=\{x\in {\mathbb{R}}^n:a_k\le x_k\le b_k\text{ for every }k<n\}$ is a compact subset of $({\mathbb{R}}^n,d_2)$ (def-metric-compactness).
2. Heine-Borel. A subset $K\subseteq {\mathbb{R}}^n$ is a compact subset of $({\mathbb{R}}^n,d_2)$ if and only if $K$ is closed in ${\mathbb{R}}^n$ (def-metric-topology) and bounded (def-metric-bounded-diameter).
3. The real line. A subset $K\subseteq \mathbb{R}$ is a compact subset of $(\mathbb{R},d_{\mathbb{R}})$, the usual metric $d_{\mathbb{R}}(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ (lem-real-line-is-a-metric-space), if and only if $K$ is closed in $\mathbb{R}$ and bounded.

No choice principle is used. The bisection below halves one coordinate at a time and takes the left half whenever the left half still fails to be finitely covered, the right half otherwise: a rule with two outcomes, decided by a property of the box, not a selection. That is the whole reason the theorem is available in ZF, while the general "complete and totally bounded implies compact" (thm-complete-and-totally-bounded-implies-compact) is not.

The hypothesis $n\ge 1$ is inherited from lem-metrics-on-rn, which defines ${\mathbb{R}}^n$ and its metrics only there; the last remark below records what happens at $n=0$.

Tight family of probability measures: A family $\mathcal{A}$ of Borel probabilities on a metric space S is tight if, for every $\epsilon >0$, there is a compact $K\subseteq S$ such that $\mu (S\setminus K)<\epsilon$ for every $\mu \in \mathcal{A}$. One K must work for the whole family. Compactness is def-metric-compactness. The empty family is tight, witnessed by the empty compact set.

Continuity and derivatives of positive-base real powers: For $a>0$, the function $x\mapsto a^x$ is continuous on $\mathbb{R}$ and $(a^x{)}^{\mathrm{\prime }}=a^x\log a\mathrm{.}$ For $\alpha \in \mathbb{R}$, the function $x\mapsto x^{\alpha }$ is continuous and differentiable on $(0,\mathrm{\infty })$, with $(x^{\alpha }{)}^{\mathrm{\prime }}=\alpha x^{\alpha -1}\mathrm{.}$

Standard normal and normal laws: Assume AC. Define $\gamma (E)={\int }_E{e}^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }dx$ for Borel E in $\mathbb{R}$. By lem-normal-density-has-total-mass-one and thm-indefinite-integral-of-a-nonnegative-function-is-a-measure, gamma is a probability measure; denote it $N(0,1)$. For $m\in \mathbb{R}$ and $\sigma \ge 0$, define $N(m,{\sigma }^2)$ as the law of $x\mapsto m+\sigma x$ on $(\mathbb{R},\mathcal{B},\gamma )$. This affine map is continuous: for $\sigma$>0 choose $\delta$=$\epsilon$/$\sigma$, and for $\sigma$=0 it is constant. Its inverse images of opens are open, so it is Borel measurable. lem-law-of-a-random-element-is-a-probability-measure makes its pushforward a probability. When $\sigma$=0, the preimage of E is all of R if m belongs to E and empty otherwise, so $N(m,0)={\delta }_m$ in def-dirac-measure.

Dominated convergence: Let $f$ and $(f_n)$ be measurable complex-valued functions such that $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single nonnegative measurable function $g$ with $\int gd\mu <+\mathrm{\infty }$. Then $f\in L^1(\mu )$, $\int \mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu ⟶0,$ and hence $\int f_{n}d\mu ⟶\int fd\mu \mathrm{.}$

Change of variables for expectation: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element, let ${\mathbb{P}}_X$ be its law, and let $g:(S,\mathrm{\Sigma })\to \mathbb{R}$ or $g:(S,\mathrm{\Sigma })\to \mathbb{C}$ be measurable.

1. If $g\ge 0$, then $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$
2. If $g(X)$ is integrable, then $g$ is integrable with respect to ${\mathbb{P}}_X$ and the same formula holds: $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

Probability laws correspond to distribution functions: Assume the Axiom of Countable Choice.

1. Let $X$ be a real random variable, let ${\mathbb{P}}_X$ be its law, and let $F_X(x)=\mathbb{P}(X\le x)$. Then $F_X$ is nondecreasing and right-continuous, satisfies ${\lim }_{x\to -\mathrm{\infty }}{F}_X(x)=0,{\lim }_{x\to +\mathrm{\infty }}{F}_X(x)=1,$ and obeys ${\mathbb{P}}_X((a,b])=F_X(b)-F_X(a)(a<b)\mathrm{.}$
2. Conversely, if $F:\mathbb{R}\to \mathbb{R}$ is nondecreasing and right-continuous with ${\lim }_{x\to -\mathrm{\infty }}F(x)=0,{\lim }_{x\to +\mathrm{\infty }}F(x)=1,$ then there is a unique Borel probability measure $\mu$ on $\mathbb{R}$ such that $\mu ((a,b])=F(b)-F(a)(a<b),$ equivalently $F(x)=\mu ((-\mathrm{\infty },x])(x\in \mathbb{R})\mathrm{.}$

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: Let $n\ge 1$, assume the Axiom of Countable Choice (def-countable-choice), and let $a_i\le b_i$ be reals for $i<n$. Write

$R^{\circ }:=\{x\in {\mathbb{R}}^n:a_i<x_i<b_i\text{ for every }i<n\},\bar{R}:=[a,b]=\{x\in {\mathbb{R}}^n:a_i\le x_i\le b_i\text{ for every }i<n\}$

(def-multidimensional-rectangle-and-volume). Then $R^{\circ }$ is open and $\bar{R}$ is closed, so both are Borel and Lebesgue measurable, and every set $R$ with $R^{\circ }\subseteq R\subseteq \bar{R}$ is Lebesgue measurable with

${\lambda }_n(R)={\prod }_{i<n}(b_i-a_i)\mathrm{.}$

In particular this covers the four one-dimensional face conventions in each coordinate — the open box, the closed box $[a,b]$, the half-open box $B(a,b)={\prod }_{i<n}(a_i,b_i]$ of def-half-open-box, and every mixture of them, in any combination of coordinates — and it gives measure $0$ to all of them whenever $a_i=b_i$ for some $i<n$. For a half-open box with infinite parameters the value is already ${\lambda }_n(B)=\mathrm{vol}(B)$ (thm-lebesgue-measure-is-a-complete-measure).

Froda's theorem: the set of discontinuities of a monotone function on an interval is at most countable, the injection into $\mathbb{N}$ being built from one fixed enumeration of the rationals by least index, so no choice principle is used: Let $I\subseteq \mathbb{R}$ be order-convex (def-interval) and let $f:I\to \mathbb{R}$ be monotone (def-monotone-function). Then the set

$D:=\{c\in I:f\text{ is discontinuous at }c\}$

(def-classification-of-discontinuities) is at most countable (def-countable).

More precisely, the proof exhibits an injection $J:D\to \mathbb{N}$ (def-injection-surjection-bijection) built from one fixed enumeration of the rationals: at a discontinuity $c$ interior to $I$ the value $J(c)$ is read off the least index of a rational lying in the gap $({\lim }_{x\to c^{-}}f(x),{\lim }_{x\to c^{+}}f(x))$, which is a nonempty open interval by thm-monotone-discontinuities-are-jumps. The map $J$ is therefore determined by $f$ and by the fixed enumeration, and no choice principle is used: least indices are canonical by thm-well-ordering-principle, and nothing anywhere in the proof is selected without being determined.

Portmanteau theorem: For Borel probabilities ${\mu }_n,\mu$ on a metric space S, the following are equivalent: (i) ${\mu }_n\Rightarrow \mu$; (ii) integrals converge for all bounded uniformly continuous real tests; (iii) ${\mathrm{lim\hspace{0.17em}sup}}_n{\mu }_n(F)\le \mu (F)$ for every closed F; (iv) ${\mathrm{lim\hspace{0.17em}inf}}_n{\mu }_n(G)\ge \mu (G)$ for every open G; (v) ${\mu }_n(A)\to \mu (A)$ for every Borel A with $\mu (\mathrm{\partial }A)=0$.

Every at most countable subset of ${\mathbb{R}}^n$ is Lebesgue null; in particular ${\lambda }_1(\mathbb{Q})=0$: Let $n\ge 1$ and assume the Axiom of Countable Choice (def-countable-choice). Every at most countable subset $E\subseteq {\mathbb{R}}^n$ (def-countable) is Lebesgue measurable with

${\lambda }_n(E)=0,$

so $E$ is a ${\lambda }_n$-null set (def-measure-null-set-and-almost-everywhere). In particular every singleton is null, and on the real line the set ${\mathbb{Q}}_{\mathbb{R}}$ of rational reals (lem-rat-embeds-dense) satisfies ${\lambda }_1({\mathbb{Q}}_{\mathbb{R}})=0$.

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

Cumulative distribution function of a real random variable: Let $X$ be a real random variable. Its cumulative distribution function is the function $F_X:\mathbb{R}\to [0,1],F_X(x):=\mathbb{P}(X\le x)={\mathbb{P}}_X((-\mathrm{\infty },x])\mathrm{.}$

The second expression is the same quantity written in terms of the law def-law-or-distribution-of-a-random-element of $X$.

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

A compact subset of a metric space is closed and bounded: Let $(X,d)$ be a metric space (def-metric-space) and let $K\subseteq X$ be a compact subset (def-metric-compactness). Then $K$ is closed in $X$ (def-metric-topology) and bounded (def-metric-bounded-diameter).

No choice principle is used: both covers below are given by a rule, and the indexed form of lem-compactness-is-intrinsic returns indices rather than sets.

The converse is false in general. A closed and bounded subset of an arbitrary metric space need not be compact (fs-closed-and-bounded-implies-compact-in-every-metric-space); it is exactly in ${\mathbb{R}}^n$ that the converse holds (thm-heine-borel-rn).

Tight family of probability measures: A family $\mathcal{A}$ of Borel probabilities on a metric space S is tight if, for every $\epsilon >0$, there is a compact $K\subseteq S$ such that $\mu (S\setminus K)<\epsilon$ for every $\mu \in \mathcal{A}$. One K must work for the whole family. Compactness is def-metric-compactness. The empty family is tight, witnessed by the empty compact set.

Portmanteau theorem: For Borel probabilities ${\mu }_n,\mu$ on a metric space S, the following are equivalent: (i) ${\mu }_n\Rightarrow \mu$; (ii) integrals converge for all bounded uniformly continuous real tests; (iii) ${\mathrm{lim\hspace{0.17em}sup}}_n{\mu }_n(F)\le \mu (F)$ for every closed F; (iv) ${\mathrm{lim\hspace{0.17em}inf}}_n{\mu }_n(G)\ge \mu (G)$ for every open G; (v) ${\mu }_n(A)\to \mu (A)$ for every Borel A with $\mu (\mathrm{\partial }A)=0$.

Continuity from below for measures: Let $(E_n{)}_{n\in \mathbb{N}}$ be an increasing sequence of measurable sets for a measure $\mu$, so $E_n\subseteq E_{n+1}$. Then

$\mu \left({\bigcup }_{n\in \mathbb{N}}{E}_n\right)={\sup }_{n\in \mathbb{N}}\mu (E_n)\mathrm{.}$

No finiteness hypothesis is required.

A uniformly integrable family: Let $(X,\mathcal{A},\mu )$ be a measure space. A family $\mathcal{F}\subseteq L^1(\mu )$ of integrable real-valued functions is uniformly integrable when ${\sup }_{f\in \mathcal{F}}{\int }_{\{\mathrm{\mid }f\mathrm{\mid }>M\}}\mathrm{\mid }f\mathrm{\mid }d\mu \to 0\text{as }M\to \mathrm{\infty }\mathrm{.}$

Equivalently, for every $\epsilon >0$ there is $M>0$ such that $f\in \mathcal{F}⟹{\int }_{\{\mathrm{\mid }f\mathrm{\mid }>M\}}\mathrm{\mid }f\mathrm{\mid }d\mu <\epsilon \mathrm{.}$

This page adopts the tail-integral definition. On finite measure spaces it is equivalent to $L^1$-boundedness plus uniform absolute continuity, proved later on this page.

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

Markov's inequality for random variables: If $X:\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ is a nonnegative random variable on a probability space and $a>0$, then $\mathbb{P}(X\ge a)\le \frac{\mathbb{E}[X]}{a}\mathrm{.}$

Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: Let $n\in \mathbb{N}$ with $n\ge 1$, let ${\mathbb{R}}^n$ be the set of functions $n\to \mathbb{R}$ and let $d_2$ be the Euclidean metric on it (lem-metrics-on-rn). Then:

1. Closed boxes are compact. For reals $a_k\le b_k$ $(k<n)$ the box $Q=\{x\in {\mathbb{R}}^n:a_k\le x_k\le b_k\text{ for every }k<n\}$ is a compact subset of $({\mathbb{R}}^n,d_2)$ (def-metric-compactness).
2. Heine-Borel. A subset $K\subseteq {\mathbb{R}}^n$ is a compact subset of $({\mathbb{R}}^n,d_2)$ if and only if $K$ is closed in ${\mathbb{R}}^n$ (def-metric-topology) and bounded (def-metric-bounded-diameter).
3. The real line. A subset $K\subseteq \mathbb{R}$ is a compact subset of $(\mathbb{R},d_{\mathbb{R}})$, the usual metric $d_{\mathbb{R}}(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ (lem-real-line-is-a-metric-space), if and only if $K$ is closed in $\mathbb{R}$ and bounded.

No choice principle is used. The bisection below halves one coordinate at a time and takes the left half whenever the left half still fails to be finitely covered, the right half otherwise: a rule with two outcomes, decided by a property of the box, not a selection. That is the whole reason the theorem is available in ZF, while the general "complete and totally bounded implies compact" (thm-complete-and-totally-bounded-implies-compact) is not.

The hypothesis $n\ge 1$ is inherited from lem-metrics-on-rn, which defines ${\mathbb{R}}^n$ and its metrics only there; the last remark below records what happens at $n=0$.

Empirical measures of iid euclidean samples converge weakly: For IID ${\mathbb{R}}^d$-valued samples $(X_i)$ with common law $\mu$ and finite $d\ge 1$, the empirical probabilities ${\widehat{\mu }}_n=n^{-1}{\sum }_{i=1}^n{\delta }_{X_i}$ converge weakly to $\mu$ almost surely on one common event.

Finite and countable subadditivity of measures: Let $\mu$ be a measure and let $(E_k{)}_{k\in \mathbb{N}}$ be measurable. Then

$\mu \left({\bigcup }_{k\in \mathbb{N}}{E}_k\right)\le {\sum }_{k=0}^{\mathrm{\infty }}\mu (E_k)\mathrm{.}$

For every $m\in \mathbb{N}$ one also has

$\mu \left({\bigcup }_{k<m}{E}_k\right)\le {\sum }_{k<m}\mu (E_k),$

including $m=0$, where both sides are $0$.