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

Definitions

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.

Properties

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

Counterexamples

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

Topology

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

where

.

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

References

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