PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Null sets are closed under countable unions and, in a complete space, under arbitrary subsets

Statement

In every measure space, a countable union of measurable null sets is measurable and null, and every measurable subset of a null set is null. If the measure space is complete, every subset of a measurable null set is measurable and null. Thus the null sets of a complete measure space form a sigma-ideal.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ .

[L1]

Countable subadditivity bounds the measure of a countable union by the sum of the member measures (Finite and countable subadditivity of measures).

[L2]

Measures are monotone under inclusion (Measures are monotone).

[L3]

A complete measure space contains every subset of every measurable null set in its sigma-algebra (Complete measure spaces).

Proof

technique · direct

1.1

If $N_{k} \in A$ and $μ (N_{k}) = 0$ for every $k$ , then $⋃_{k} N_{k}$ is measurable and [L1] gives $0 \leq μ (⋃_{k} N_{k}) \leq \sum_{k} 0 = 0$ .

givenL1

1.2

If $S \in A$ and $S \subseteq N$ for a measurable null set $N$ , then $0 \leq μ (S) \leq μ (N) = 0$ .

givenL2

2.1

If the space is complete and $S \subseteq N$ for a measurable null set $N$ , then [L3] first makes $S$ measurable and step 1.2 makes it null.

step 1.2L3

3.1

Step 1.1 gives closure under countable unions, and step 2.1 gives closure under arbitrary subsets in a complete space, including the empty union and the empty subset.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 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

G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)