DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-28

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A tight family of integrable functions

Definition

Let $(X, A, μ)$ be a measure space. A family $F \subseteq L^{1} (μ)$ of integrable real-valued functions is tight when for every $ε > 0$ there is a measurable set $E \in A$ with $μ (E) < + \infty$ such that

$\sup_{f \in F} \int_{X ∖ E} ∣ f ∣ d μ < ε .$

On a finite measure space every family is automatically tight by taking $E = X$ , so tightness matters only on infinite spaces.

Depends on

Used by

Vitali convergence theorem on finite and sigma-finite measure spaces Theorem

Dependency tree · two levels

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Sources

H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th ed., Section 5.1 (standard reference, not scraped)