False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-31

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FALSE: the epsilon-delta condition characterises absolute continuity for every measure

Statement

False claim: For every signed or complex measure $ν$ , absolute continuity $ν ≪ μ$ is equivalent to the epsilon-delta small-set condition.

Facts & Assumptions

Given: The measure $ν (E) = \int_{E} x^{- 1} χ_{(0, 1)} (x) d λ (x)$ .

[L1]

This nonnegative density defines a measure (The measure with density $f$ relative to $μ$ ), and the integral over every Lebesgue-null set vanishes (A nonnegative integral over a null set vanishes), so the measure is absolutely continuous with respect to Lebesgue measure by Absolute continuity of a signed or complex measure with respect to a positive measure.

[L2]

The valid theorem requires finiteness. (For finite signed or complex measures, absolute continuity is equivalent to the epsilon-delta small-set condition)

Refutation

technique · direct

1.1

By [L1], the measure $ν$ satisfies $ν ≪ λ$ .

L1given

2.1

With $ε = 1$ , every $δ > 0$ fails: for $E = (0, δ / 2)$ one has $λ (E) < δ$ but $ν (E) = \int_{0}^{δ / 2} \frac{d x}{x} = + \infty > 1.$ Thus the epsilon-delta condition does not follow from absolute continuity in this sigma-finite but nonfinite case.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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, Exercise 14 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Proposition 13.2 discussion (standard reference, not scraped)