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

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Continuous mapping theorem

Statement

Let S,T be metric spaces, $g : S \to T$ measurable, and $μ_{n} \Rightarrow μ$ . If the discontinuity set $D_{g}$ is $μ$ -null, then $g_{*} μ_{n} \Rightarrow g_{*} μ$ . Consequently $X_{n} \Rightarrow X$ implies $g (X_{n}) \Rightarrow g (X)$ whenever $P_{X} (D_{g}) = 0$ .

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]

Laws commute with measurable maps: Let $X : (Ω, F, P) \to (S, Σ)$ be a random element, and let $g : (S, Σ) \to (T, T)$ be measurable. Then $g \circ X$ is a random element and for every $B \in T$ , $P_{g \circ X} (B) = P_{X} (g^{- 1} (B)) .$

Proof

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

1.1givenalgebra

For $r \geq 1$ let $U_{r}$ be the union of all open subsets V of S such that the diameter of g(V) is less than 1/r. The continuity set is $⋂_{r} U_{r}$ : continuity at x supplies such a neighborhood by making all image points within 1/(3r) of g(x); conversely a neighborhood with image diameter below $ε$ forces $d (g (y), g (x)) < ε$ there. Thus $D_{g}$ is Borel.

1.2F1

If F is closed in T and x lies outside $g^{- 1} (F) \cup D_{g}$ , continuity at x and the open complement of F give a neighborhood disjoint from $g^{- 1} (F)$ . Hence $\overline{g^{- 1} (F)} \subseteq g^{- 1} (F) \cup D_{g}$ . Applying F1 to this closed preimage closure gives ${lim sup}_{n} μ_{n} (g^{- 1} (F)) \leq μ (\overline{g^{- 1} (F)}) \leq μ (g^{- 1} (F))$ .

2.1F1F2step 1.2∎

The inequality in step 1.2 is the closed-set bound for the pushforward probabilities, so F1 gives their weak convergence. F2 identifies these pushforwards with the laws of g( $X_{n}$ ) and g(X), proving the random-element formulation.

Depends on

Used by

Dependency tree · two levels

15 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 3.2; local closed-preimage argument (standard reference, not scraped)