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CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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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.
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Holder's inequality for random variables

Statement

Let p,q∈[1,∞] be conjugate exponents. If X and Y are real random variables in the spaces named by the corresponding clause of Holder's inequality for integrals, including the endpoint cases, then E[∣XY∣]≤∥X∥p∥Y∥q.

In particular, XY is integrable.

Facts & Assumptions

Given: Real random variables X,Y and conjugate exponents p,q as in the Statement.

[L1]

Expectation is integration against the probability measure (Expectation of a nonnegative or integrable random variable).

[L2]

Holder's integral inequality, including the endpoint cases, holds on every measure space (Holder's inequality for integrals, including the endpoint cases).

Proof

technique · direct
1.1

Apply [L2] to the measure space (Ω,F,P). Rewriting the left-hand side with [L1] gives E[∣XY∣]=∫∣XY∣ dP≤∥X∥p∥Y∥q.

L1L2
2.1

The same theorem [L2] already states that the right-hand side is finite in every allowed case, so XY is integrable.

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

14 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