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For finite signed or complex measures, absolute continuity is equivalent to the epsilon-delta small-set condition

Statement

Assume the Axiom of Choice. Let $μ$ be a sigma-finite positive measure and let $ν$ be a finite signed measure or a finite complex measure on the same measurable space. Then the following are equivalent:

$ν ≪ μ$ ;
for every $ε > 0$ there exists $δ > 0$ such that $μ (E) < δ ⟹ ∣ ν (E) ∣ < ε (E \in A) .$

Facts & Assumptions

Given: The Axiom of Choice, a finite signed or finite complex measure $ν$ and a sigma-finite positive measure $μ$ .

[L1]

Under the stated Axiom of Choice, if $ν ≪ μ$ , then $∣ ν ∣ (E) = \int_{E} ∣ d ν / d μ ∣ d μ$ for every measurable $E$ (The Radon-Nikodym derivative as an almost-everywhere equivalence class, The total variation of an absolutely continuous signed or complex measure has density the absolute value of the Radon-Nikodym derivative, The Axiom of Choice).

[L2]

Absolute continuity of the integral gives the $ε$ - $δ$ estimate for integrable absolute values. (Absolute continuity of the integral)

[L3]

For signed or complex measures, $∣ ν ∣ ≪ μ$ is equivalent to $ν ≪ μ$ , and always $∣ ν (E) ∣ \leq ∣ ν ∣ (E)$ . (For signed and complex measures, absolute continuity is equivalent for the measure, its Jordan or real-imaginary parts, and its total variation, The total variation |nu|(E) from countable measurable partitions)

Proof

technique · direct

1.1L1L2L3choose

Assume $ν ≪ μ$ and let $ε > 0$ . By [L1], the function $∣ d ν / d μ ∣$ is integrable, so [L2] yields $δ > 0$ such that $μ (E) < δ$ implies $∣ ν ∣ (E) = \int_{E} ∣ \frac{d ν}{d μ} ∣ d μ < ε .$ Then [L3] gives $∣ ν (E) ∣ < ε$ . This proves condition 2.

1.2givenalgebra

Assume condition 2. If $μ (E) = 0$ , then $μ (E) < δ$ for every $δ > 0$ , so condition 2 forces $∣ ν (E) ∣ < ε$ for every $ε > 0$ . Hence $ν (E) = 0$ , and therefore $ν ≪ μ$ .

2.1step 1.1step 1.2∎

Step 1.1 proves $(1) \Rightarrow (2)$ and step 1.2 proves $(2) \Rightarrow (1)$ .

Depends on

Used by

x⁻¹dλ on (0,1) shows finiteness is needed in the epsilon-delta criterion Counterexample
FALSE: the epsilon-delta condition characterises absolute continuity for every measure False statement

Dependency tree · two levels

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Sources

Richard F. Bass, Real Analysis for Graduate Students, Proposition 13.2 (standard reference, not scraped)
John K. Hunter, Measure Theory, Proposition 6.25 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Exercise 14 (standard reference, not scraped)