False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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FALSE: every finitely additive nonnegative set function on a sigma-algebra is a measure

Statement

False claim. Every finitely additive nonnegative set function on a sigma-algebra is countably additive and therefore is a measure.

Facts & Assumptions

Given: The power-set sigma-algebra $P (N)$ .

[L1]

Finite additivity requires additivity on disjoint pairs and value $0$ at the empty set (Finitely additive nonnegative set functions).

[L2]

A measure must be countably additive on disjoint measurable sequences (Measures on sigma-algebras).

[L3]

Finite sets are those equinumerous with a natural number (Finite, countably infinite, countable, uncountable).

Refutation

technique · direct

1.1

Define $m (A) = 0$ when $A \subseteq N$ is finite and $m (A) = + \infty$ when $A$ is infinite; in particular $m (\emptyset) = 0$ .

givenL3

2.1

If disjoint $A, B$ have finite union, then both are finite and $m (A \cup B) = 0 = 0 + 0$ ; if their union is infinite, at least one is infinite and both sides of $m (A \cup B) = m (A) + m (B)$ are $+ \infty$ . Thus $m$ is finitely additive.

step 1.1L1

2.2

The singletons ${k}$ are disjoint, each has value $0$ , and their union is $N$ , which has value $+ \infty$ .

step 1.1L3

3.1

Step 2.2 violates countable additivity, so the finitely additive set function of step 2.1 is not a measure and the claim is false.

step 2.1step 2.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)