 Convergence in measure

A sequence of real measurable functions converges in measure to $f$ when the displayed condition holds for every positive threshold. It controls the measure of the exceptional set at each index, rather than requiring eventual closeness on each sample path. On a probability space it is exactly convergence in probability. On a finite measure space, almost-everywhere convergence implies convergence in measure by continuity of measure on the decreasing tail events. The converse need not hold for the full sequence, but convergence in measure always admits an almost-everywhere convergent subsequence.

Convergence in measure is a concept from measure theory, which is a branch of mathematics dealing with the formalization of notions like size, length, and area. It is particularly important in the study of sequences of measurable functions.