CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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Convergence in $L^{p}$ implies convergence in measure

Statement

Let $1 \leq p \leq \infty$ , let $u_{n}, u \in L^{p} (μ)$ , and choose measurable representatives $f_{n}, f$ . If $∥ u_{n} - u ∥_{p} \to 0$ , then $f_{n} \to f$ in measure.

Facts & Assumptions

Given: Representatives $f_{n}, f$ of classes $u_{n}, u \in L^{p} (μ)$ with $∥ u_{n} - u ∥_{p} \to 0$ .

[L1]

Convergence in measure means the bad-set measures $μ ({∣ f_{n} - f ∣ > ε})$ tend to $0$ for every $ε > 0$ (Convergence in measure).

[L2]

The class norm agrees with the representative norm (The $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$ ).

[L3]

Chebyshev-Markov gives $μ ({h \geq t}) \leq t^{- 1} \int h d μ$ for nonnegative measurable $h$ (Chebyshev-Markov inequality for the integral).

[L4]

If $∥ g ∥_{\infty} < \infty$ , then $∣ g ∣ \leq ∥ g ∥_{\infty}$ almost everywhere (The essential supremum is attained as the least essential bound).

Proof

Proof technique: For $1 \leq p < i n f i n i t y$ , apply Chebyshev-Markov to $∣ f_{n} - f ∣^{p}$ . For $p = i n f i n i t y$ , the essential-supremum bound makes the bad set null for all large $n$ .

1.1

Assume $1 \leq p < \infty$ and fix $ε > 0$ . Apply [L3] to [L2, L3, given, algebra] $h : = ∣ f_{n} - f ∣^{p}$ and $t : = ε^{p}$ : $μ ({∣ f_{n} - f ∣ > ε}) = μ ({∣ f_{n} - f ∣^{p} > ε^{p}}) \leq ε^{- p} \int ∣ f_{n} - f ∣^{p} d μ = ε^{- p} ∥ u_{n} - u ∥_{p}^{p} .$ By [L2], the right-hand side tends to $0$ .

1.2

Assume $p = \infty$ and fix $ε > 0$ . Because $∥ u_{n} - u ∥_{\infty} \to 0$ , [L2, L4, given] for all large $n$ one has $∥ u_{n} - u ∥_{\infty} < ε$ . Then [L4] gives $∣ f_{n} - f ∣ \leq ε$ almost everywhere, so $μ ({∣ f_{n} - f ∣ > ε}) = 0.$ Hence the bad-set measures are eventually $0$ .

2.1

Step 1.1 proves the finite- $p$ case and step 1.2 proves the [step 1.1, step 1.2, L1] $p = \infty$ case, so [L1] gives convergence in measure. ∎

Depends on

Used by

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Sources

John K. Hunter, Measure Theory, Chapter 13 and Chapter 15 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Section 2.5 (standard reference, not scraped)

Convergence in LpLp implies convergence in measure