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

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.

Expectation depends only on the almost-everywhere class

Statement

If $X$ and $Y$ are integrable real or complex random variables on one probability space and $X = Y$ almost surely, then $E [X] = E [Y] .$

Thus expectation is a function of the $L^{1}$ -equivalence class.

Facts & Assumptions

Given: Integrable random variables $X, Y$ on one probability space.

[L1]

Expectation is the Lebesgue integral with respect to the underlying probability measure (Expectation of a nonnegative or integrable random variable).

[L2]

Two integrable functions are equal almost everywhere exactly when all of their integrals over measurable sets agree (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

Proof

technique · direct

1.1

Since $X = Y$ almost surely and both are integrable, [L2] applied to the measurable set $Ω$ gives $\int_{Ω} X d P = \int_{Ω} Y d P .$

L2given

2.1

Rewriting the two integrals as expectations by [L1] yields $E [X] = E [Y]$ .

step 1.1L1∎

Depends on

Used by

Equality almost surely is not pointwise equality Counterexample

Dependency tree · two levels

8 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 1.5 (standard reference, not scraped)