LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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Finite random variables are measurable

Statement

Let $(Ω, w)$ be a finite probability space, and regard it as the probability space $(Ω, P (Ω), P_{w})$ from Finite probability spaces are exactly finite full-power-set probability spaces. Then every function $X : Ω \to R$ is a real random variable.

In particular, the published finite definition Real random variables on finite probability spaces and their finite distributions is exactly the measure-theoretic definition on that full-power-set probability space.

Facts & Assumptions

Given: A finite probability space $(Ω, w)$ and a function $X : Ω \to R$ .

[L1]

The theorem on finite probability spaces identifies $(Ω, w)$ with a probability measure on $(Ω, P (Ω))$ (Finite probability spaces are exactly finite full-power-set probability spaces).

[L2]

A real random variable is a measurable map from the sample-space sigma-algebra to $(R, B (R))$ (Random elements and real random variables).

[L3]

On a finite probability space, a real random variable is simply a function $Ω \to R$ (Real random variables on finite probability spaces and their finite distributions).

Proof

technique · direct

1.1

By [L1], every subset of $Ω$ is measurable. Hence for every Borel set $B \subseteq R$ , the preimage $X^{- 1} (B)$ is a subset of $Ω$ , so it lies in $P (Ω)$ . Therefore $X$ is measurable.

L1L2

2.1

Step 1.1 proves that every finite random variable in the sense of [L3] is a real random variable in the sense of [L2], so the two notions agree exactly on finite full-power-set probability spaces.

step 1.1L2L3∎

Depends on

Used by

Expectation agrees with the published finite weighted sum Corollary

Dependency tree · two levels

11 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

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.3 (standard reference, not scraped)
J. R. Norris, Probability and Measure, Section 2.1 (standard reference, not scraped)