TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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A signed or complex measure that is both absolutely continuous and singular with respect to the same positive measure is zero

Statement

Let $μ$ be a positive measure and let $ν$ be a signed measure or a complex measure on the same measurable space. If $ν ≪ μ$ and $ν ⊥ μ$ , then $ν = 0$ .

Facts & Assumptions

Given: A positive measure $μ$ and a signed or complex measure $ν$ with $ν ≪ μ$ and $ν ⊥ μ$ .

[L1]

Mutual singularity provides measurable sets $P, N$ with $P ⊔ N = X$ , every measurable subset of $N$ having $μ$ -value $0$ , and every measurable subset of $P$ having $ν$ -value $0$ . (Mutual singularity for signed or complex measures)

[L2]

Absolute continuity means that every measurable $μ$ -null set has $ν$ -value $0$ . (Absolute continuity of a signed or complex measure with respect to a positive measure)

Proof

technique · direct

1.1L1choose

Choose measurable sets $P, N$ as in [L1]. Then every measurable subset of $P$ has $ν$ -value $0$ .

1.2L1L2given

Let $A \in A$ . Because $A \cap N \subseteq N$ , [L1] gives $μ (A \cap N) = 0$ , so [L2] yields $ν (A \cap N) = 0$ ; because $A \cap P \subseteq P$ , [L1] also gives $ν (A \cap P) = 0$ .

2.1step 1.1step 1.2algebra∎

The partition $A = (A \cap P) ⊔ (A \cap N)$ therefore gives $ν (A) = 0$ . Since $A$ was arbitrary, $ν = 0$ .

Depends on

Used by

A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density Theorem

Dependency tree · two levels

6 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

Sheldon Axler, Measure, Integration & Real Analysis, 9.34 (standard reference, not scraped)
John K. Hunter, Measure Theory, Lemma 6.26 (standard reference, not scraped)