Tightness from a uniform moment bound

${\mathbb{R}}^n$ 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 $(x^{\alpha }{)}^{\mathrm{\prime }}=\alpha x^{\alpha -1}\mathrm{.}$