A nontight sequence with no probability law subsequence limit

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.