ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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.

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The measure $δ_{0} + λ ⁣ ↾_{[0, 1]}$ splits into discrete and absolutely continuous parts

Example

Assume the Axiom of Countable Choice. Let $μ : = δ_{0} + λ ⁣ ↾_{[0, 1]} .$ Then in the three-part decomposition of Every finite Borel measure on R has a unique absolutely continuous, discrete, and singular-continuous decomposition one has $μ_{d} = δ_{0}, μ_{a c} = λ ⁣ ↾_{[0, 1]}, μ_{s c} = 0.$

Facts & Assumptions

Given: The finite Borel measure $μ = δ_{0} + λ ⁣ ↾_{[0, 1]}$ .

[L1]

Dirac measure is a finite Borel measure concentrated at one point. (The Dirac set function at a point, A Dirac set function is a probability measure)

[A1]

The restriction $λ ⁣ ↾_{[0, 1]}$ is absolutely continuous with respect to $λ$ .

[L3]

Every finite Borel measure on $R$ has a unique absolutely continuous, discrete, and singular-continuous decomposition. (Every finite Borel measure on R has a unique absolutely continuous, discrete, and singular-continuous decomposition)

Verification

technique · direct

1.1

By [L1], $δ_{0}$ is discrete. By [A1], $λ ⁣ ↾_{[0, 1]}$ is absolutely continuous with respect to $λ$ . Their sum is $μ$ , and neither summand has an atomless singular part.

L1A1given

2.1

Therefore the displayed decomposition has exactly the form required by [L3], and uniqueness there forces $μ_{d} = δ_{0}$ , $μ_{a c} = λ ⁣ ↾_{[0, 1]}$ , and $μ_{s c} = 0$ .

step 1.1L3∎

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The measure δ0+λ ⁣↾[0,1]δ0​+λ↾[0,1]​ splits into discrete and absolutely continuous parts