# Convergence in measure

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

## Definitions

Let $f,f_{n}\ (n\in \mathbb {N} ):X\to \mathbb {R}$ 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,

$\lim _{n\to \infty }\mu (\{x\in X:|f(x)-f_{n}(x)|\geq \varepsilon \})=0$ ,
$\lim _{n\to \infty }\mu (\{x\in F:|f(x)-f_{n}(x)|\geq \varepsilon \})=0$ .

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 $\in$ 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 $X=\mathbb {R}$ , μ be Lebesgue measure, and f the constant function with value zero.

(The first five terms of which are $\chi _{\left[0,1\right]},\;\chi _{\left[0,{\frac {1}{2}}\right]},\;\chi _{\left[{\frac {1}{2}},1\right]},\;\chi _{\left[0,{\frac {1}{4}}\right]},\;\chi _{\left[{\frac {1}{4}},{\frac {1}{2}}\right]}$ ) 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

$\{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},$ where

$\rho _{F}(f,g)=\int _{F}\min\{|f-g|,1\}\,d\mu$ .

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 $G\subset X$ of finite measure and $\varepsilon >0$ there exists F in the family such that $\mu (G\setminus F)<\varepsilon .$ When $\mu (X)<\infty$ , we may consider only one metric $\rho _{X}$ , 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.