Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.


Let be measurable functions on a measure space . The sequence is said to converge globally in measure to if for every ,


and to converge locally in measure to if for every and every with ,


On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.


Throughout, f and fn (n N) are measurable functions XR.


Let , μ be Lebesgue measure, and f the constant function with value zero.


There is a topology, called the topology of (local) convergence in measure, on the collection of measurable functions from X such that local convergence in measure corresponds to convergence on that topology. This topology is defined by the family of pseudometrics



In general, one may restrict oneself to some subfamily of sets F (instead of all possible subsets of finite measure). It suffices that for each of finite measure and there exists F in the family such that When , we may consider only one metric , so the topology of convergence in finite measure is metrizable. If is an arbitrary measure finite or not, then

still defines a metric that generates the global convergence in measure.[1]

Because this topology is generated by a family of pseudometrics, it is uniformizable. Working with uniform structures instead of topologies allows us to formulate uniform properties such as Cauchyness.

See also


  1. ^ Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007