# 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 $f,f_{n}\ (n\in \mathbb {N} ):X\to \mathbb {R}$ be measurable functions on a measure space $(X,\Sigma ,\mu )$ . The sequence $f_{n}$ is said to converge globally in measure to $f$ if for every $\epsilon >0$ ,

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

and to converge locally in measure to $f$ if for every $\epsilon >0$ and every $F\in \Sigma$ with $\mu (F)<\infty$ ,

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

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, however, $\mu (X)<\infty$ or, more generally, if f and all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
• 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.
• Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure.[clarification needed]
• 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.[clarification needed]
• 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.[clarification needed]
• 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 sequence $f_{n}=\chi _{[n,\infty )}$ converges to f locally in measure, but does not converge to f globally in measure.
• The sequence $f_{n}=\chi _{[{\frac {j}{2^{k}}},{\frac {j+1}{2^{k}}}]}$ where $k=\lfloor \log _{2}n\rfloor$ and $j=n-2^{k}$ (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 globally in measure; but for no x does fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.

• The sequence $f_{n}=n\chi _{\left[0,{\frac {1}{n}}\right]}$ converges to f almost everywhere and globally in measure, but not in the p-norm for any $p\geq 1$ .

## 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. If $\mu$ is an arbitrary measure finite or not, then

$d(f,g):=\inf \limits _{\delta >0}\mu (\{|f-g|\geq \delta \})+\delta$ 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.