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

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FALSE: measures are additive on arbitrary countable unions

Statement

False claim. For every measure and every measurable sequence $(E_{k})$ , whether or not it is pairwise disjoint,

$μ (⋃_{k} E_{k}) = \sum_{k} μ (E_{k}) .$

Facts & Assumptions

Given: The one-point measurable space $X = {x}$ with sigma-algebra $P (X)$ and its Dirac probability $δ_{x}$ .

[L1]

Countable additivity in the definition of a measure applies only to pairwise disjoint sequences (Measures on sigma-algebras).

[L2]

The Dirac set function assigns $1$ to measurable sets containing $x$ and $0$ otherwise (The Dirac set function at a point), and it is a probability measure (A Dirac set function is a probability measure).

[L3]

A nonnegative extended series starts at index $0$ and is the supremum of its partial sums (Series in the nonnegative extended real line).

Refutation

technique · direct

1.1

Define $E_{0} = E_{1} = {x}$ and $E_{k} = \emptyset$ for $k \geq 2$ . Then $⋃_{k} E_{k} = {x}$ .

given

2.1

The union has Dirac measure $1$ , while the term values are $1, 1, 0, 0, \dots$ , whose nonnegative extended sum is $2$ .

step 1.1L2L3algebra

3.1

Since $1 \neq 2$ , the claimed equality fails; the repeated set at indices $0$ and $1$ pinpoints the absent disjointness hypothesis in [L1].

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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, Definition 2.54 (standard reference, not scraped)