Convergence in measure


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 ' to if for every
and to ' 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, and are measurable functions.
  • Global convergence in measure implies local convergence in measure. The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general.
  • If, however, or, more generally, if and all the vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
  • If is σ-finite and converges to in measure, there is a subsequence converging to almost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
  • If is -finite, converges to locally in measure if and only if every subsequence has in turn a subsequence that converges to almost everywhere.
  • In particular, if converges to almost everywhere, then converges to locally in measure. The converse is false.
  • Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by convergence in measure.
  • If is -finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by convergence in measure.
  • If and μ is Lebesgue measure, there are sequences of step functions and of continuous functions converging globally in measure to.
  • If and are in Lp(μ) for some and converges to in the -norm, then converges to globally in measure. The converse is false.
  • If converges to in measure and converges to in measure then converges to in measure. Additionally, if the measure space is finite, also converges to.

Counterexamples

Let, be Lebesgue measure, and the constant function with value zero.
  • The sequence converges to locally in measure, but does not converge to globally in measure.
  • The sequence
  • The sequence

Topology

There is a topology, called the topology of 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. 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.
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.