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A uniformly integrable family
Definition
Let be a measure space. A family of integrable real-valued functions is uniformly integrable when
Equivalently, for every there is such that
This page adopts the tail-integral definition. On finite measure spaces it is equivalent to -boundedness plus uniform absolute continuity, proved later on this page.
Depends on
Used by
- A uniformly integrable family need not admit a single integrable majorant Example
- FALSE: uniform integrability implies domination by one integrable function False statement
- Dominated families are uniformly integrable Proposition
- On a finite measure space, uniform integrability is equivalent to L¹-boundedness plus uniform absolute continuity Theorem
- Vitali convergence theorem on finite and sigma-finite measure spaces Theorem
Dependency tree · two levels
5 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
- Terence Tao, 245A Notes 4: Modes of convergence, Exercise 22 (standard reference, not scraped)
- Richard F. Bass, Real Analysis for Graduate Students, Exercise 7.21 (standard reference, not scraped)