Convergence in measure is either of two distinct mathematical concepts both of which generalizethe concept ofconvergence in probability.

Let${\displaystyle f,f_n(n\in \mathbb{N}):X\to \mathbb{R}}$ bemeasurable functions on ameasure space${\displaystyle (X,\mathrm{\Sigma },\mu ).}$ The sequence${\displaystyle f_n}$ is said toconverge globally in measure to${\displaystyle f}$ if for every${\displaystyle \epsilon >0,}$$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mu (\{x\in X:|f(x)-f_n(x)|\ge \epsilon \})=0,}$$and toconverge locally in measure to${\displaystyle f}$ if for every${\displaystyle \epsilon >0}$ and every${\displaystyle F\in \mathrm{\Sigma }}$ with$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mu (\{x\in F:|f(x)-f_n(x)|\ge \epsilon \})=0.}$$

On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure^[1]: 2.2.3 or local convergence in measure, depending on the author.

Throughout,${\displaystyle f}$ and${\displaystyle f_n}$ (${\displaystyle n\in \mathbb{N}}$) are measurable functions${\displaystyle X\to \mathbb{R}}$.

• 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${\displaystyle f}$ and all the${\displaystyle f_n}$ vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
• If${\displaystyle \mu }$ isσ-finite and (f_n) converges (locally or globally) to${\displaystyle f}$ in measure, there is a subsequence converging to${\displaystyle f}$almost everywhere.^[1]: 2.2.5 The assumption ofσ-finiteness is not necessary in the case of global convergence in measure.
• If${\displaystyle \mu }$ is${\displaystyle \sigma }$-finite,${\displaystyle (f_n)}$ converges to${\displaystyle f}$ locally in measureif and only if every subsequence has in turn a subsequence that converges to${\displaystyle f}$ almost everywhere.
• In particular, if${\displaystyle (f_n)}$ converges to${\displaystyle f}$ almost everywhere, then${\displaystyle (f_n)}$ converges to${\displaystyle f}$ locally in measure. The converse is false.
• Fatou's lemma and themonotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure.
• If${\displaystyle \mu }$ is${\displaystyle \sigma }$-finite, Lebesgue'sdominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.^[1]: 2.8.6
• If${\displaystyle X=[a,b]\subseteq \mathbb{R}}$ andμ isLebesgue measure, there are sequences${\displaystyle (g_n)}$ of step functions and${\displaystyle (h_n)}$ of continuous functions converging globally in measure to${\displaystyle f}$.
• If${\displaystyle f}$ and${\displaystyle f_n}$ are inL^p(μ) for some${\displaystyle p>0}$ and${\displaystyle (f_n)}$ converges to${\displaystyle f}$ in the${\displaystyle p}$-norm, then${\displaystyle (f_n)}$ converges to${\displaystyle f}$ globally in measure. The converse is false.
• If${\displaystyle f_n}$ converges to${\displaystyle f}$ in measure and${\displaystyle g_n}$ converges to${\displaystyle g}$ in measure then${\displaystyle f_n+g_n}$ converges to${\displaystyle f+g}$ in measure. Additionally, if the measure space is finite,${\displaystyle f_n{g}_n}$ also converges to${\displaystyle fg}$.

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

• The sequence${\displaystyle f_n={\chi }_{[n,\mathrm{\infty })}}$ converges to${\displaystyle f}$ locally in measure, but does not converge to${\displaystyle f}$ globally in measure.
• The sequence

${\displaystyle f_n={\chi }_{\left[\frac{j}{2^k},\frac{j+1}{2^k}\right]},}$

where${\displaystyle k=\lfloor {\log }_{2}n\rfloor }$ and${\displaystyle j=n-2^k}$, the first five terms of which are

${\displaystyle {\chi }_{\left[0,1\right]},{\chi }_{\left[0,\frac{1}{2}\right]},{\chi }_{\left[\frac{1}{2},1\right]},{\chi }_{\left[0,\frac{1}{4}\right]},{\chi }_{\left[\frac{1}{4},\frac{1}{2}\right]},}$

converges to${\displaystyle 0}$ globally in measure; but for no${\displaystyle x}$ does${\displaystyle f_n(x)}$ converge to zero. Hence${\displaystyle (f_n)}$ fails to converge to${\displaystyle f}$ almost everywhere.^[1]: 2.2.4

• The sequence

${\displaystyle f_n=n{\chi }_{\left[0,\frac{1}{n}\right]}}$

converges to${\displaystyle f}$ almost everywhere and globally in measure, but not in the${\displaystyle p}$-norm for any${\displaystyle p\ge 1}$.

There is atopology, called thetopology of (local) convergence in measure, on the collection of measurable functions fromX such that local convergence in measure corresponds to convergence on that topology.This topology is defined by the family ofpseudometricswhere$${\displaystyle {\rho }_F(f,g)={\int }_{F}min\{|f-g|,1\}d\mu .}$$In general, one may restrict oneself to some subfamily of setsF (instead of all possible subsets of finite measure). It suffices that for each${\displaystyle G\subset X}$ of finite measure and${\displaystyle \epsilon >0}$ there existsF in the family such that When, we may consider only one metric${\displaystyle {\rho }_X}$, so the topology of convergence in finite measure is metrizable. If${\displaystyle \mu }$ is an arbitrary measure finite or not, then$${\displaystyle d(f,g):=\underset{\delta >0}{inf}\mu (\{|f-g|\ge \delta \})+\delta }$$still defines a metric that generates the global convergence in measure.^[2]

Because this topology is generated by a family of pseudometrics, it isuniformizable.Working with uniform structures instead of topologies allows us to formulateuniform properties such asCauchyness.

• Convergence space

• 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.

• Measure theory
• Convergence (mathematics)
• Lp spaces

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