CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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Two four-point probability measures agree on a generating family that is not a pi-system

Statement refuted

Agreement on a family that merely generates the sigma-algebra does not determine a measure. Intersection closure in the pi-system uniqueness theorem is essential.

Facts & Assumptions

Given: The four-point set $X = {N W, N E, S W, S E}$ and the family $G$ consisting of $X$ , the north and south rows, and the west and east columns.

[L1]

A probability measure is a measure with total mass $1$ (Probability measures and probability spaces).

[L2]

The generated sigma-algebra is the smallest sigma-algebra containing the generating family (The sigma-algebra generated by a family of sets).

[L3]

A pi-system is closed under intersections (Pi-systems), and the uniqueness theorem requires agreement on a generating pi-system with a finite-measure exhaustion (Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system).

Counterexample

technique · direct

1.1

Give $μ$ masses $(1 / 2, 0, 0, 1 / 2)$ at $(N W, N E, S W, S E)$ and give $ν$ masses $(0, 1 / 2, 1 / 2, 0)$ . Finite summation over subsets defines probability measures on $P (X)$ .

givenL1

1.2

Intersecting rows with columns produces every singleton, so $σ_{X} (G) = P (X)$ ; a singleton intersection is absent from $G$ , so $G$ is not a pi-system. The constant sequence $X, X, \dots$ is an increasing finite-measure exhaustion lying in $G$ .

givenL1L2L3

2.1

Every row and column contains exactly one $μ$ -atom and one $ν$ -atom of mass $1 / 2$ , and both measures give $X$ mass $1$ , so the measures agree on $G$ .

givenstep 1.1algebra

3.1

The measures are unequal because $μ ({N W}) = 1 / 2$ and $ν ({N W}) = 0$ , despite steps 2.1 and 1.2. The family satisfies the generating and finite-measure exhaustion hypotheses and violates the pi-system hypothesis of [L3].

step 1.1step 2.1step 1.2L3∎

Depends on

Used by

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Sources

D. Pollard, A User's Guide to Measure Theoretic Probability, §10, Example 42 (standard reference, not scraped)