Fatou's lemma

Fatou's lemma states that the integral of the pointwise lower limit of nonnegative measurable functions is at most the lower limit of their integrals.

Pointwise convergence: A sequence of functions converges pointwise when its values converge at every point of the domain. Fatou's lemma uses the pointwise lower limit, which records the eventual lower behavior at each point.
Measure space: A set equipped with a sigma-algebra and a countably additive measure. The functions and integrals in Fatou's lemma are defined on a measure space.
Weak convergence of measures: Convergence of measures tested against bounded continuous functions. Portmanteau inequalities use Fatou-type arguments to bound integrals of nonnegative lower semicontinuous functions.
Monotone convergence theorem: A theorem stating that integrals commute with increasing limits of nonnegative measurable functions. Unlike Fatou's inequality, it gives equality when the functions increase pointwise.
Pierre Fatou: A French mathematician known for work in complex analysis, dynamical systems, and integration theory. The lemma bears Fatou's name and reflects his work on passing from sequences to integral bounds.
Limit inferior: The limit inferior of a sequence is the limit of the infima of its tails. The lemma compares the integral of pointwise limit inferiors with the limit inferior of integrals.
Nonnegative function: A real-valued or extended-real-valued function whose values are at least zero. Nonnegativity is the key hypothesis that makes Fatou's inequality hold.
Portmanteau theorem: A collection of equivalent conditions characterizing weak convergence of probability measures. Its lower-bound condition for lower semicontinuous functions is an integral form of Fatou's principle.
Dominated convergence theorem: A theorem allowing limits to pass through integrals under an integrable dominating function. It concludes convergence of integrals, while Fatou's lemma gives only a one-sided bound.
Henri Lebesgue: A French mathematician who developed the measure-theoretic integral now bearing his name. Fatou's lemma belongs to the integration framework created by Lebesgue's theory.

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Almost-everywhere convergenceRelated: It supplies an integral bound under almost-everywhere convergence without guaranteeing convergence of integrals.