Uniform integrability

Uniform integrability is a condition on a family of integrable functions that makes their integrals over sets of small measure uniformly small. It rules out mass concentrating in increasingly tall, narrow spikes.

Vitali convergence theorem: A theorem characterizing convergence in integral through convergence in measure and uniform integrability. It turns uniform integrability into the key condition for upgrading convergence in measure to convergence in mean.
Convergence in measure: Convergence in which the measure of the set where errors exceed any fixed positive threshold tends to zero. Together with uniform integrability, it yields convergence in mean under the Vitali theorem.
Lebesgue integral: An integral defined by measuring function values across measurable subsets of a measure space. Uniform integrability constrains how these integrals behave across a family of functions.
Boundedness in L1: A family of integrable functions whose L1 norms have a common finite upper bound. An L1 bound controls total mass but alone permits mass to concentrate on tiny sets.
de la Vallée-Poussin criterion: A criterion stating that a family is uniformly integrable when a suitable convex growth function has uniformly bounded integrals. It provides a practical tail-growth test equivalent to uniform integrability.
Weak convergence of measures: Convergence of measures defined by their integrals against bounded continuous test functions. Uniform integrability controls tails when weak convergence is used to pass to expectations.
Measure space: A set equipped with a collection of measurable subsets and a measure assigning them sizes. The small-set definition depends on measuring subsets of the functions' domain.
Dominated convergence theorem: A theorem allowing limits to pass through integrals when functions converge almost everywhere and share an integrable bound. Domination guarantees uniform integrability, while uniform integrability can apply without a single integrable dominator.
Uniform absolute continuity of integrals: The property that integrals over measurable sets tend uniformly to zero as the sets' measures tend to zero. This is the defining small-set condition for uniform integrability.
Prokhorov's theorem: A theorem characterizing tightness as relative compactness for probability measures on a complete separable metric space. Uniform integrability complements tightness when seeking convergence of expectations, not just distributions.

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