Almost-everywhere convergence

A sequence of measurable functions converges almost everywhere if it converges pointwise except on a set of measure zero. The exceptional set may depend on the sequence.

Pointwise convergence: A sequence of functions converges pointwise when its values converge at every point in the domain. Almost-everywhere convergence weakens pointwise convergence by allowing failure on a null set.
Null set: A measurable set of measure zero, which may be nonempty and may be uncountable. The points where convergence fails must fit inside such a set.
Strong law of large numbers: A theorem stating that sample averages converge almost surely to the expected value under suitable conditions. Almost-sure convergence is precisely almost-everywhere convergence on a probability space.
Typewriter sequence: A sequence of indicator functions that converges pointwise almost everywhere but not in measure. It shows almost-everywhere convergence need not imply convergence in measure on an infinite-measure space.
Convergence in measure: A sequence converges in measure when the measure of the set where its error exceeds any fixed threshold tends to zero. Unlike almost-everywhere convergence, this condition controls error-set sizes separately at each stage.
Limsup of sets: The limsup of a sequence of sets consists of points belonging to infinitely many of those sets. Error sets' limsup describes points where a sequence fails to converge to its proposed limit.
Fourier series: A representation of a periodic function as a sum of sinusoidal terms. Convergence of Fourier partial sums is often studied almost everywhere rather than at every point.
Rademacher functions: A sequence of functions on the unit interval taking values plus or minus one on alternating dyadic intervals. They converge to zero in measure but fail to converge almost everywhere.
Convergence in probability: Random variables converge in probability when the probability of an error exceeding any fixed threshold tends to zero. On a probability space, this is convergence in measure for the underlying probability measure.
Borel–Cantelli lemma: A pair of measure-theoretic results relating sums of event probabilities to membership in infinitely many events. Its first lemma often proves almost-everywhere convergence by making error events summable.

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