CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Weak limits are unique

Statement

Bounded continuous real tests determine Borel probability measures on any metric space. In particular, weak limits are unique.

Facts & Assumptions

[F1]

Portmanteau theorem: For Borel probabilities $μ_{n}, μ$ on a metric space S, the following are equivalent: (i) $μ_{n} \Rightarrow μ$ ; (ii) integrals converge for all bounded uniformly continuous real tests; (iii) ${lim sup}_{n} μ_{n} (F) \leq μ (F)$ for every closed F; (iv) ${lim inf}_{n} μ_{n} (G) \geq μ (G)$ for every open G; (v) $μ_{n} (A) \to μ (A)$ for every Borel A with $μ (\partial A) = 0$ .

[F2]

Dynkin's pi-lambda theorem: Let $P$ be a $π$ -system on $X$ . Then $λ_{X} (P) = σ_{X} (P)$ . Consequently, if $D$ is any lambda-system on $X$ with $P \subseteq D$ , then $σ_{X} (P) \subseteq D$ .

Proof

Given: The objects, hypotheses and definitions in the statement. Its conclusions are to be established below.

1.1

If $μ$ and $ν$ have equal integrals of every bounded continuous test, the constant sequence $μ$ converges weakly to $ν$ . F1 gives $μ (F) \leq ν (F)$ for every closed F. Reverse the roles to get equality.

2.1

The class of Borel sets on which the two probabilities agree contains S, is closed under complements and disjoint countable unions, and contains the closed sets by step 1.1. Closed sets form a $π$ -system generating the Borel $σ$ -algebra; F2 therefore gives equality on all Borel sets. If a sequence has two weak limits, uniqueness of each numerical integral limit gives the hypothesis of step 1.1, so those limits agree.

F2step 1.1∎

Depends on

Used by

Interval realization from refining small diameter partitions Lemma

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

van Gaans, Theorem 4.1, definiteness argument, pp. 9–10 (standard reference, not scraped)