Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A noncylinder path functional may fail to be measurable

Statement refuted

Every set-defined path functional on the uncountable Bernoulli path space is measurable for its cylinder sigma-algebra.

Counterexample

Assume AC. For uncountable I, let A={x∈{0,1}I:{i:xi=1} is uncountable} and put f=1A.

Given: AC, an uncountable index set I, and its Bernoulli cylinder space.

1.1

If A were cylinder-measurable, the countable-coordinate lemma would give a countable J⊆I determining it.

given
2.1

Choose i∈I∖J. The all-zero path and the path with ones on I∖J agree on J, but only the latter belongs to A; this contradicts determination by J.

step 1.1
3.1

Hence A is not cylinder-measurable and f is not a real random variable on the canonical cylinder probability space.

step 2.1∎

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Sources