Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Uniqueness of the RMK representing measure among Radon measures

Statement

If two Radon measures μ and ν on an LCH space satisfy ∫f dμ=∫f dν for every f∈Cc(X), then μ=ν on B(X).

Facts & Assumptions

Given: μ,ν are Radon and agree on all Cc integrals.

[L1]

LCH cutoffs exist between compact and open sets. (LCH Urysohn cutoff)

Proof

technique · direct
1.1

If K⊆U with K compact and U open, choose f by [L1]. Then μ(K)≤∫f dμ=∫f dν≤ν(U). Taking the infimum over U⊇K gives μ(K)≤ν(K) by outer regularity; symmetry gives equality on compact sets.

L1
2.1

Inner regularity on open sets now gives μ(U)=ν(U) for every open U. Outer regularity on Borel sets then gives μ(E)=ν(E) for every Borel E, including infinite values.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

13 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