TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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.

Dominated convergence in $L^{p}$

Statement

Let $1 \leq p < \infty$ . If $X_{n} \to X$ almost surely and $∣ X_{n} ∣ \leq Y$ almost surely for every $n$ , where $Y \in L^{p} (P)$ , then $X_{n} \to X$ in $L^{p}$ .

Facts & Assumptions

Given: $1 \leq p < \infty$ , $X_{n} \to X$ almost surely, and $∣ X_{n} ∣ \leq Y$ almost surely with $Y \in L^{p}$ .

[L1]

Dominated convergence gives convergence of integrals under one integrable majorant (Dominated convergence).

[L2]

$L^{p}$ convergence is convergence of the $p$ th absolute moments of the difference ( $L^{p}$ convergence for random variables).

Proof

technique · direct

1.1givenalgebra

Intersect the countably many full-measure events on which $∣ X_{n} ∣ \leq Y$ with the full-measure convergence event. Outside the resulting null set, $X_{n} \to X$ and $∣ X_{n} ∣ \leq Y$ for every $n$ , so $∣ X ∣ \leq Y$ there. Since $Y \in L^{p}$ , these bounds also show that every $X_{n}$ and $X$ belongs to $L^{p}$ . Moreover, $∣ X_{n} - X ∣^{p} \to 0$ almost everywhere and $∣ X_{n} - X ∣^{p} \leq (2 ∣ Y ∣)^{p}$ .

2.1step 1.1L1L2∎

Since $(2 ∣ Y ∣)^{p}$ is integrable, [L1] applied to step 1.1 yields $E ∣ X_{n} - X ∣^{p} \to 0$ . By [L2] and the established $L^{p}$ membership, this is $X_{n} \to X$ in $L^{p}$ .

Depends on

Used by

Localized Ito integral Theorem

Dependency tree · two levels

15 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., Theorem 1.5.8 (standard reference, not scraped)

Dominated convergence in LpLp