Real cdf and bounded continuous definitions agree

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 $\mu (\mathrm{\partial }A)=0$.

Convergence in distribution for real random variables: For real random variables $(X_n)$ and $X$, write $X_n\Rightarrow X$, or $X_n\to X$ in distribution, when $F_{X_n}(x)⟶F_X(x)$ at every continuity point $x$ of $F_X$. Here $F_X$ is the CDF from def-cumulative-distribution-function-of-a-random-variable and continuity points are those of def-atom-and-continuity-point-of-a-law.

Continuity from above when one set has finite measure: Let $(E_n{)}_{n\in \mathbb{N}}$ be a decreasing sequence of measurable sets for a measure $\mu$. If $\mu (E_{n_0})<+\mathrm{\infty }$ for some $n_0$, then

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

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.

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 $n=0$.