# 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 (*X*,Σ,*μ*). The sequence (*f*_{n}) 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 *f*_{n} (*n* **N**) are measurable functions *X* → **R**.

- 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 all the
*f*_{n}vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears. - If
*μ*is*σ*-finite and (*f*_{n}) 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, (*f*_{n}) 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 (
*f*_{n}) converges to*f*almost everywhere, then (*f*_{n}) converges to*f*locally in measure. The converse is false. - Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure.
- If
*μ*is*σ*-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure. - If
*X*= [*a*,*b*] ⊆**R**and*μ*is Lebesgue measure, there are sequences (*g*_{n}) of step functions and (*h*_{n}) of continuous functions converging globally in measure to*f*. - If
*f*and*f*_{n}(*n*∈**N**) are in*L*^{p}(*μ*) for some*p*> 0 and (*f*_{n}) converges to*f*in the*p*-norm, then (*f*_{n}) converges to*f*globally in measure. The converse is false. - If
*f*_{n}converges to*f*in measure and*g*_{n}converges to*g*in measure then*f*_{n}+*g*_{n}converges to*f*+*g*in measure. Additionally, if the measure space is finite,*f*_{n}*g*_{n}also converges to*fg*.

## Counterexamples

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

- The sequence converges to
*f*locally in measure, but does not converge to*f*globally in measure. - The sequence where and

(The first five terms of which are )
converges to *0* locally in measure; but for no *x* does *f _{n}(x)* converge to zero.
Hence

*(f*fails to converge to

_{n})*f*almost everywhere.

- The sequence converges to
*f*almost everywhere (hence also locally in measure), but not in the*p*-norm for any .

## 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.

## References

- ↑ Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007

- 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.