Convergence in measure

Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.

Let ${\displaystyle f,f_{n}\ (n\in \mathbb {N} ):X\to \mathbb {R} }${\displaystyle f,f_{n}\ (n\in \mathbb {N} ):X\to \mathbb {R} } be measurable functions on a measure space ${\displaystyle (X,\Sigma ,\mu ).}${\displaystyle (X,\Sigma ,\mu ).} The sequence ${\displaystyle f_{n}}${\displaystyle f_{n}} is said to converge globally in measure to ${\displaystyle f}${\displaystyle f} if for every ${\displaystyle \varepsilon >0,}${\displaystyle \varepsilon >0,} $${\displaystyle \lim _{n\to \infty }\mu (\{x\in X:|f(x)-f_{n}(x)|\geq \varepsilon \})=0,}$${\displaystyle \lim _{n\to \infty }\mu (\{x\in X:|f(x)-f_{n}(x)|\geq \varepsilon \})=0,} and to converge locally in measure to ${\displaystyle f}${\displaystyle f} if for every ${\displaystyle \varepsilon >0}${\displaystyle \varepsilon >0} and every ${\displaystyle F\in \Sigma }${\displaystyle F\in \Sigma } with ${\displaystyle \mu (F)<\infty ,}${\displaystyle \mu (F)<\infty ,} $${\displaystyle \lim _{n\to \infty }\mu (\{x\in F:|f(x)-f_{n}(x)|\geq \varepsilon \})=0.}$${\displaystyle \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^{[1] : 2.2.3} or local convergence in measure, depending on the author.

Throughout, ${\displaystyle f}${\displaystyle f} and ${\displaystyle f_{n}}${\displaystyle f_{n}} (${\displaystyle n\in \mathbb {N} }${\displaystyle n\in \mathbb {N} }) are measurable functions ${\displaystyle X\to \mathbb {R} }${\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, ${\displaystyle \mu (X)<\infty }${\displaystyle \mu (X)<\infty } or, more generally, if ${\displaystyle f}${\displaystyle f} and all the ${\displaystyle f_{n}}${\displaystyle f_{n}} vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
• If ${\displaystyle \mu }${\displaystyle \mu } is σ-finite and (f_n) converges (locally or globally) to ${\displaystyle f}${\displaystyle f} in measure, there is a subsequence converging to ${\displaystyle f}${\displaystyle f} almost everywhere.^{[1] : 2.2.5} The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
• If ${\displaystyle \mu }${\displaystyle \mu } is ${\displaystyle \sigma }${\displaystyle \sigma }-finite, ${\displaystyle (f_{n})}${\displaystyle (f_{n})} converges to ${\displaystyle f}${\displaystyle f} locally in measure if and only if every subsequence has in turn a subsequence that converges to ${\displaystyle f}${\displaystyle f} almost everywhere.
• In particular, if ${\displaystyle (f_{n})}${\displaystyle (f_{n})} converges to ${\displaystyle f}${\displaystyle f} almost everywhere, then ${\displaystyle (f_{n})}${\displaystyle (f_{n})} converges to ${\displaystyle f}${\displaystyle 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 ${\displaystyle \mu }${\displaystyle \mu } is ${\displaystyle \sigma }${\displaystyle \sigma }-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.^{[1] : 2.8.6}
• If ${\displaystyle X=[a,b]\subseteq \mathbb {R} }${\displaystyle X=[a,b]\subseteq \mathbb {R} } and μ is Lebesgue measure, there are sequences ${\displaystyle (g_{n})}${\displaystyle (g_{n})} of step functions and ${\displaystyle (h_{n})}${\displaystyle (h_{n})} of continuous functions converging globally in measure to ${\displaystyle f}${\displaystyle f}.
• If ${\displaystyle f}${\displaystyle f} and ${\displaystyle f_{n}}${\displaystyle f_{n}} are in L^p(μ) for some ${\displaystyle p>0}${\displaystyle p>0} and ${\displaystyle (f_{n})}${\displaystyle (f_{n})} converges to ${\displaystyle f}${\displaystyle f} in the ${\displaystyle p}${\displaystyle p}-norm, then ${\displaystyle (f_{n})}${\displaystyle (f_{n})} converges to ${\displaystyle f}${\displaystyle f} globally in measure. The converse is false.
• If ${\displaystyle f_{n}}${\displaystyle f_{n}} converges to ${\displaystyle f}${\displaystyle f} in measure and ${\displaystyle g_{n}}${\displaystyle g_{n}} converges to ${\displaystyle g}${\displaystyle g} in measure then ${\displaystyle f_{n}+g_{n}}${\displaystyle f_{n}+g_{n}} converges to ${\displaystyle f+g}${\displaystyle f+g} in measure. Additionally, if the measure space is finite, ${\displaystyle f_{n}g_{n}}${\displaystyle f_{n}g_{n}} also converges to ${\displaystyle fg}${\displaystyle fg}.

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

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

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

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

${\displaystyle \chi _{\left[0,1\right]},\;\chi _{\left[0,{\frac {1}{2}}\right]},\;\chi _{\left[{\frac {1}{2}},1\right]},\;\chi _{\left[0,{\frac {1}{4}}\right]},\;\chi _{\left[{\frac {1}{4}},{\frac {1}{2}}\right]},}${\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 ${\displaystyle 0}${\displaystyle 0} globally in measure; but for no ${\displaystyle x}${\displaystyle x} does ${\displaystyle f_{n}(x)}${\displaystyle f_{n}(x)} converge to zero. Hence ${\displaystyle (f_{n})}${\displaystyle (f_{n})} fails to converge to ${\displaystyle f}${\displaystyle f} almost everywhere.^{[1] : 2.2.4}

• The sequence

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

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

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 $${\displaystyle \{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},}$${\displaystyle \{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},} where $${\displaystyle \rho _{F}(f,g)=\int _{F}\min\{|f-g|,1\},円d\mu .}$${\displaystyle \rho _{F}(f,g)=\int _{F}\min\{|f-g|,1\},円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 ${\displaystyle G\subset X}${\displaystyle G\subset X} of finite measure and ${\displaystyle \varepsilon >0}${\displaystyle \varepsilon >0} there exists F in the family such that ${\displaystyle \mu (G\setminus F)<\varepsilon .}${\displaystyle \mu (G\setminus F)<\varepsilon .} When ${\displaystyle \mu (X)<\infty }${\displaystyle \mu (X)<\infty }, we may consider only one metric ${\displaystyle \rho _{X}}${\displaystyle \rho _{X}}, so the topology of convergence in finite measure is metrizable. If ${\displaystyle \mu }${\displaystyle \mu } is an arbitrary measure finite or not, then $${\displaystyle d(f,g):=\inf \limits _{\delta >0}\mu (\{|f-g|\geq \delta \})+\delta }$${\displaystyle d(f,g):=\inf \limits _{\delta >0}\mu (\{|f-g|\geq \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 is uniformizable. Working with uniform structures instead of topologies allows us to formulate uniform properties such as Cauchyness.

• 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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