• Images
• Videos
• Documents
• Books
• Articles

Definitions

Let f , f n   ( n ∈ N ) : X → R {\displaystyle f,f_{n}\ (n\in \mathbb {N} ):X\to \mathbb {R} } be measurable functions on a measure space ( X , Σ , μ ) . {\displaystyle (X,\Sigma ,\mu ).} The sequence f n {\displaystyle f_{n}} is said to converge globally in measure to f {\displaystyle f} if for every ε > 0 , {\displaystyle \varepsilon >0,} lim n → ∞ μ ( { x ∈ X : | f ( x ) − f n ( x ) | ≥ ε } ) = 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 f {\displaystyle f} if for every ε > 0 {\displaystyle \varepsilon >0} and every F ∈ Σ {\displaystyle F\in \Sigma } with μ ( F ) < ∞ , {\displaystyle \mu (F)<\infty ,} lim n → ∞ μ ( { x ∈ F : | f ( x ) − f n ( x ) | ≥ ε } ) = 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 or local convergence in measure, depending on the author.

Properties

Throughout, f and fn (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, μ ( X ) < ∞ {\displaystyle \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.
• 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 (gn) of step functions and (hn) of continuous functions converging globally in measure to f.
• If f and fn (n ∈ N) 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 = R {\displaystyle X=\mathbb {R} } μ be Lebesgue measure, and f the constant function with value zero.

• The sequence f n = χ [ n , ∞ ) {\displaystyle f_{n}=\chi _{[n,\infty )}} converges to f locally in measure, but does not converge to f globally in measure.
• The sequence f n = χ [ j 2 k , j + 1 2 k ] {\displaystyle f_{n}=\chi _{\left[{\frac {j}{2^{k}}},{\frac {j+1}{2^{k}}}\right]}} where k = ⌊ log 2 ⁡ n ⌋ {\displaystyle k=\lfloor \log _{2}n\rfloor } and j = n − 2 k {\displaystyle j=n-2^{k}} (The first five terms of which are χ [ 0 , 1 ] , χ [ 0 , 1 2 ] , χ [ 1 2 , 1 ] , χ [ 0 , 1 4 ] , χ [ 1 4 , 1 2 ] {\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 fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.

• The sequence f n = n χ [ 0 , 1 n ] {\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 p ≥ 1 {\displaystyle 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 { ρ F : F ∈ Σ ,   μ ( F ) < ∞ } , {\displaystyle \{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},} where ρ F ( f , g ) = ∫ F min { | f − g | , 1 } d μ . {\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 G ⊂ X {\displaystyle G\subset X} of finite measure and ε > 0 {\displaystyle \varepsilon >0} there exists F in the family such that μ ( G ∖ F ) < ε . {\displaystyle \mu (G\setminus F)<\varepsilon .} When μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } , we may consider only one metric ρ X {\displaystyle \rho _{X}} , so the topology of convergence in finite measure is metrizable. If μ {\displaystyle \mu } is an arbitrary measure finite or not, then d ( f , g ) := inf δ > 0 μ ( { | f − g | ≥ δ } ) + δ {\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.

See also

• 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.

References

1. Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007 ↩