TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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.

Levy continuity theorem forward direction

Statement

If Borel probability laws $μ_{n} \Rightarrow μ$ on $R$ , then $φ_{μ_{n}} (t) \to φ_{μ} (t)$ for every real $t$ .

Facts & Assumptions

Given: The hypotheses and conventions in the statement.

[F1]

Weak convergence tests every bounded continuous real function. Weak convergence of borel probability measures.

[F2]

The complex integral is componentwise. Characteristic function of a real random variable.

Proof

technique · direct

1.1

Fix $t \in R$ . The real functions $x \mapsto \cos (t x)$ and $x \mapsto \sin (t x)$ are continuous and bounded by one, so weak convergence gives convergence of each of their integrals against $μ_{n}$ to the corresponding integral against $μ$ .

2.1

By the componentwise definition, the cosine integrals are the real parts of the characteristic functions and the sine integrals their imaginary parts. Combining the two convergences gives the asserted complex limit. At t=0 these two integral sequences are constantly one and zero respectively. Since t was arbitrary the result holds at every frequency.

step 1.1F2∎

Depends on

Used by

Characteristic function criterion for weak convergence Corollary
Cramer wold device Theorem
Levy continuity theorem converse Theorem

Dependency tree · two levels

9 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

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
Norris, Probability and Measure (standard reference, not scraped)