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

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Assuming countable choice, counting measure is sigma-finite exactly on countable sets

Example

Assume the Axiom of Countable Choice. For a set $X$ , counting measure on $(X, P (X))$ is sigma-finite if and only if $X$ is at most countable. In particular, counting measure on $R$ is not sigma-finite.

Facts & Assumptions

Given: A set $X$ , its counting measure $#$ , and the Axiom of Countable Choice.

[L1]

Counting measure assigns finite measure exactly to finite sets (Counting measure on an arbitrary set) and is a measure (Counting measure is a measure).

[L2]

Sigma-finiteness means that the whole space is a countable union of measurable finite-measure sets (Finite, sigma-finite, and semifinite measures).

[L3]

Under countable choice, a countable union of at most countable sets is at most countable (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Countable unions of at most countable sets, assuming ${A C}_{ω}$ ).

[L4]

The real line is uncountable ( $R$ is uncountable (Cantor's nested intervals, 1874)).

Verification

technique · direct

1.1

For the forward implication, suppose counting measure on $X$ is sigma-finite. By [L2], $X = ⋃_{n} E_{n}$ with $# (E_{n}) < + \infty$ , so every $E_{n}$ is finite by [L1] and $X$ is at most countable by [L3].

givenL1L2L3

1.2

For the reverse implication when $X$ is finite, the constant cover $E_{n} = X$ has finite counting measure; when $X$ is countably infinite, a bijection $e : N \to X$ gives finite initial sets $E_{n} = {e (k) : k < n}$ with union $X$ . The empty set is covered by the constant empty sequence.

givenL1L2

2.1

Steps 1.1 and 1.2 prove the equivalence. Since $R$ is uncountable by [L4], the forward implication shows that its counting measure is not sigma-finite.

step 1.1step 1.2L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

33 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

S. Axler, Measure, Integration & Real Analysis, §2C (standard reference, not scraped)
G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)