# 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 $\varepsilon >0,$ $\lim _{n\to \infty }\mu (\{x\in X:|f(x)-f_{n}(x)|\geq \varepsilon \})=0,$ and to converge locally in measure to $f$ if for every $\varepsilon >0$ and every $F\in \Sigma$ with $\mu (F)<\infty ,$ $\lim _{n\to \infty }\mu (\{x\in F:|f(x)-f_{n}(x)|\geq \varepsilon \})=0.$
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, f and f$_n$ (n $\in$ 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, $\mu (X)<\infty$ or, more generally, if f and 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 $X=\mathbb {R}$ 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 $f_{n}=\chi _{\left[{\frac {j}{2^{k}}},{\frac {j+1}{2^{k}}}\right]}$ 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 f$_n$(x) converge to zero. Hence (f$_n$) 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.