DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The finite product of finite probability spaces

Definition

Let $((Ω_{i}, w_{i}))_{i \in I}$ be a finite indexed family of finite probability spaces. Their product probability space has outcome set $Ω : = \prod_{i \in I} Ω_{i}$ and assigns $ω = (ω_{i})_{i \in I}$ the weight $w (ω) : = \prod_{i \in I} w_{i} (ω_{i}) .$ For $I = \emptyset$ , the product outcome set is the singleton containing the empty function and its weight is the empty product $1$ . Normalization and independence of coordinate events are proved in Product weights normalize, and coordinate events are mutually independent ↗.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Definitions 4.3-4.5 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4.1 (standard reference, not scraped)