ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Dirac laws converge weakly exactly when their points converge

Example

On a metric space S, $δ_{x_{n}} \Rightarrow δ_{x}$ if and only if $x_{n} \to x$ . For example, on the real line $δ_{1 / (n + 1)} \Rightarrow δ_{0}$ .

Facts & Assumptions

[F1]

Weak convergence of borel probability measures: For Borel probability measures $μ_{n}, μ$ on a metric space S, write $μ_{n} \Rightarrow μ$ if $\int f d μ_{n} \to \int f d μ$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int ∣ f ∣ d μ \leq ∥ f ∥_{\infty} μ (S) < \infty$ , so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.

Verification

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

1.1

A unit point mass is a probability: among disjoint sets at most one contains its point, so the indicator formula is countably additive. Its integral of a bounded measurable f is f at that point, first for simple functions and then by bounded approximation. If $x_{n} \to x$ , continuity gives $\int f d δ_{x_{n}} = f (x_{n}) \to f (x) = \int f d δ_{x}$ for each bounded continuous f. By F1 this is weak convergence.

2.1

Conversely test weak convergence with $f (y) = \min (1, d (y, x))$ , a bounded continuous function. Its limiting integral is f(x)=0, so min(1,d( $x_{n}$ ,x))->0; for $ε$ <1 this forces d( $x_{n}$ ,x)< $ε$ eventually. Thus x_n->x. For the displayed example this distance is $∣ 1 / (n + 1) - 0 ∣ = 1 / (n + 1) \to 0$ .

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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, §9, Dirac embedding (standard reference, not scraped)