False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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FALSE: every finitely additive nonnegative function on an algebra extends to a measure

Statement

Every finitely additive function from an algebra of subsets to $[0, + \infty]$ that vanishes at the empty set extends to a measure on the generated sigma-algebra.

Facts & Assumptions

Given: The power-set algebra $P (N)$ and the function $λ (A) = 0$ for finite $A$ and $λ (A) = + \infty$ for infinite $A$ .

[F1]

An algebra of subsets of $X$ is a subfamily of $P (X)$ containing $\emptyset$ and closed under complements and finite unions. (Algebras of subsets)

[F2]

A measure vanishes at the empty set and is countably additive on every pairwise disjoint sequence in its sigma-algebra, beginning at index $0$ and allowing $+ \infty$ . (Measures on sigma-algebras)

Refutation

technique · direct

1.1

The function vanishes at $\emptyset$ . If $A$ and $B$ are disjoint and both finite, their union is finite and $0 = 0 + 0$ ; if their union is infinite, at least one of $A, B$ is infinite, so both $λ (A \cup B)$ and $λ (A) + λ (B)$ are $+ \infty$ . Thus $λ$ is finitely additive on the algebra [F1].

F1algebra

2.1

The algebra is already the sigma-algebra $P (N)$ , so any extension must equal $λ$ there. But the disjoint singleton sequence has union $N$ , while $λ (N) = + \infty$ and $\sum_{k = 0}^{\infty} λ ({k}) = 0$ , contradicting [F2].

step 1.1F2algebra∎

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

T. Tao, An Introduction to Measure Theory, Section 1.7.2 and Exercise 1.7.6 (standard reference, not scraped)