Convergence in measure explained

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

Definitions

Let

f, f _{n (n} \in N): X \to R be measurable functions on a measure space (X, \Sigma, \mu). The sequence f _{n} is said to to f if for every \varepsilon > 0 , $$\backslash lim\_\; \backslash mu(\backslash )\; =\; 0,$$and to to f if for every \varepsilon> 0 and every F \in \Sigma with\mu (F) < inf ty, $$\backslash lim\_\; \backslash mu(\backslash )\; =\; 0.$$

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)< inf ty 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.

Counterexamples

Let

X = \Reals 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[ (j)/(, (j +1)/(2 ^{k}) \right] 2 ^{k})} 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 , (12 \right])/()} , \chi _{\left[ (12 , 1 \right])/()} , \chi _{\left[ 0 , (14 \right])/()} , \chi _{\left[ (14 , (12 \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 , (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$$\backslash ,$$where$$\backslash rho\_F(f,g)\; =\; \backslash int\_F\; \backslash min\backslash$$

,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) < inf ty , 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$$d(f,g)\; :=\; \backslash inf\backslash limits\_\; \backslash mu(\backslash$$

\geq\delta\

) + \deltastill 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 asCauchyness.

See also

• Convergence space

References

• 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.

Notes and References

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