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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The law of a random element is a probability measure

Statement

Let $X : (Ω, F, P) \to (S, Σ)$ be a random element. Then $P_{X}$ is a probability measure on $(S, Σ)$ .

Facts & Assumptions

Given: A random element $X : (Ω, F, P) \to (S, Σ)$ .

[L1]

The law is defined by $P_{X} (B) = P (X^{- 1} (B))$ (Law or distribution of a random element).

[L2]

A random element is measurable, so measurable target sets have measurable preimages (Random elements and real random variables).

[L3]

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

Proof

technique · direct

1.1

By [L2], every $B \in Σ$ has $X^{- 1} (B) \in F$ , so [L1] is well defined. Also $X^{- 1} (\emptyset) = \emptyset$ and $X^{- 1} (S) = Ω$ , so $P_{X} (\emptyset) = 0, P_{X} (S) = P (Ω) = 1.$

L1L2L3

1.2

If $(B_{n})$ is a pairwise disjoint sequence in $Σ$ , then the preimages $X^{- 1} (B_{n})$ are pairwise disjoint and $X^{- 1} (⋃_{n} B_{n}) = ⋃_{n} X^{- 1} (B_{n}) .$ Therefore $P_{X} (⋃_{n} B_{n}) = P (⋃_{n} X^{- 1} (B_{n})) = \sum_{n = 0}^{\infty} P (X^{- 1} (B_{n})) = \sum_{n = 0}^{\infty} P_{X} (B_{n}) .$

L1L2L3

2.1

Steps 1.1 and 1.2 show that $P_{X}$ is a measure of total mass $1$ , hence a probability measure by [L3].

step 1.1step 1.2L3∎

Depends on

Used by

Laws commute with measurable maps Lemma
Change of variables for expectation Theorem
Probability laws correspond to distribution functions Theorem

Dependency tree · two levels

7 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 1.2 (standard reference, not scraped)
S. R. S. Varadhan, Probability Theory, Section 1.6 (standard reference, not scraped)