Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30
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FALSE: agreement on a generating pi-system always determines a signed measure

Statement

False claim. If two signed measures agree on a generating pi-system, then they agree on the whole generated sigma-algebra.

Facts & Assumptions

Given: The two-point space X={0,1} with sigma-algebra P(X), and the family P:={{0}}.

[L1]

A pi-system is a nonempty family closed under binary intersections. (Pi-systems)

[L2]

A signed measure is any countably additive set function with the at-most-one-infinite-sign convention. (A signed measure is countably additive and takes at most one infinite value)

Refutation

technique · direct
1.1

The family P={{0}} is a pi-system by [L1], and [L1, L2] σ(P)=P(X) because complements and unions recover {1} and X. Define signed measures μ(A):=0,ν(A):=1A(1). They agree on the generating pi-system element {0}.

2.1

However μ({1})=0 while ν({1})=1, so the signed measures are not [L2, step 1.1] ∎ equal on the generated sigma-algebra. Therefore the claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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