Convergence in measure

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Convergence in measure can refer to two distinct mathematical concepts which both generalize the concept of convergence in probability.

Definitions

Let be measurable functions on a measure space (X,Σ,μ). The sequence (fn) is said to converge globally in measure to f if for every ε > 0,

,

and to converge locally in measure to f if for every ε > 0 and every with ,

.

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.

  • 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 μ is σ-finite and (fn) converges (locally or globally) to f in measure, there is a subsequence converging to f almost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
  • If μ is σ-finite, (fn) converges to f locally in measure if and only if every subsequence has in turn a subsequence that converges to f almost everywhere.
  • In particular, if (fn) converges to f almost everywhere, then (fn) converges to f locally in measure. The converse is false.
  • If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (gn) of step functions and (hn) of continuous functions converging globally in measure to f. Template:Clarify
  • If f and fn (nN) are in Lp(μ) for some p > 0 and (fn) converges to f in the p-norm, then (fn) converges to f globally in measure. The converse is false.
  • If fn converges to f in measure and gn converges to g in measure then fn + gn converges to f + g in measure. Additionally, if the measure space is finite, fngn also converges to fg.

Counterexamples

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

(The first five terms of which are ) converges to 0 locally in measure; but for no x does fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.

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.

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.

References

  • D.H. Fremlin, 2000. Measure Theory. Torres Fremlin.
  • H.L. Royden, 1988. Real Analysis. Prentice Hall.
  • G.B. Folland 1999, Section 2.4. Real Analysis. John Wiley & Sons.