Convergence in measure

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

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

Throughout, f and f_n (n ${\displaystyle \in }$  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, ${\displaystyle \mu (X)<\infty }$  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_ng_n also converges to fg.

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

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

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 $${\displaystyle \{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},}$$  where $${\displaystyle \rho _{F}(f,g)=\int _{F}\min\{|f-g|,1\}\,d\mu .}$$  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 ${\displaystyle G\subset X}$  of finite measure and ${\displaystyle \varepsilon >0}$  there exists F in the family such that ${\displaystyle \mu (G\setminus F)<\varepsilon .}$  When ${\displaystyle \mu (X)<\infty }$ , 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 $${\displaystyle d(f,g):=\inf \limits _{\delta >0}\mu (\{|f-g|\geq \delta \})+\delta }$$  still defines a metric that generates the global convergence in measure.^[1]

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.