Convergence in measure

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

Definitions

Let $f,f_{n}\ (n\in {\mathbb N}):X\to {\mathbb R}$ be measurable functions on a measure space (X,Σ,μ). The sequence (f_n) is said to converge globally in measure to f if for every ε > 0,

\lim _{{n\to \infty }}\mu (\{x\in X:|f(x)-f_{n}(x)|\geq \varepsilon \})=0

and to converge locally in measure to f if for every ε > 0 and every $F\in \Sigma$ with $\mu (F)<\infty$ ,

\lim _{{n\to \infty }}\mu (\{x\in F:|f(x)-f_{n}(x)|\geq \varepsilon \})=0

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, $\mu (X)<\infty$ or, more generally, if 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 $X={\mathbb R}$ , 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 $f_{n}=\chi _{{[{\frac {j}{2^{k}}},{\frac {j+1}{2^{k}}}]}}$ where $k=\lfloor \log _{2}n\rfloor$ and $j=n-2^{k}$

(The first five terms of which are $\chi _{{\left[0,1\right]}},\;\chi _{{\left[0,{\frac 12}\right]}},\;\chi _{{\left[{\frac 12},1\right]}},\;\chi _{{\left[0,{\frac 14}\right]}},\;\chi _{{\left[{\frac 14},{\frac 12}\right]}}$ ) converges to 0 locally 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 $f_{n}=n\chi _{{\left[0,{\frac 1n}\right]}}$ converges to f almost everywhere (hence also locally 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

\{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},

where

\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 $G\subset X$ of finite measure and $\varepsilon >0$ there exists F in the family such that $\mu (G\setminus F)<\varepsilon .$ When $\mu (X)<\infty$ , 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) := \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.

References

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

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.

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