TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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$L^{q}$ convergence implies $L^{p}$ convergence on a probability space

Statement

If $1 \leq p < q \leq \infty$ and $X_{n} \to X$ in $L^{q}$ on a probability space, then $X_{n} \to X$ in $L^{p}$ .

Facts & Assumptions

Given: $1 \leq p < q \leq \infty$ and $X_{n} \to X$ in $L^{q}$ .

[L1]

A probability measure has total mass one (Probability measures and probability spaces).

[L2]

On a finite measure space, $∥ f ∥_{p} \leq μ (Ω)^{1 / p - 1 / q} ∥ f ∥_{q}$ for $p < q$ , including the $q = \infty$ case (Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$ ).

Proof

technique · direct

1.1

Apply [L2] to $X_{n} - X$ and use [L1] to obtain the following bound. [L1, L2] $∥ X_{n} - X ∥_{p} \leq ∥ X_{n} - X ∥_{q}$ .

L1L2

2.1

The right-hand side in step 1.1 tends to zero by the given $L^{q}$ convergence, so the left-hand side does too. This is $L^{p}$ convergence.

step 1.1∎

Depends on

Used by

Complete convergence implication diagram Remark

Dependency tree · two levels

12 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

S. Roch, Lecture 3: Modes of convergence, Theorem 3.12 (standard reference, not scraped)

LqLq convergence implies LpLp convergence on a probability space