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 [math]\displaystyle{ f, f_n\ (n \in \mathbb N): X \to \mathbb R }[/math] be measurable functions on a measure space [math]\displaystyle{ (X, \Sigma, \mu) }[/math]. The sequence [math]\displaystyle{ f_n }[/math] is said to converge globally in measure to [math]\displaystyle{ f }[/math] if for every [math]\displaystyle{ \varepsilon \gt 0 }[/math],
- [math]\displaystyle{ \lim_{n\to\infty} \mu(\{x \in X: |f(x)-f_n(x)|\geq \varepsilon\}) = 0 }[/math],
and to converge locally in measure to [math]\displaystyle{ f }[/math] if for every [math]\displaystyle{ \varepsilon\gt 0 }[/math] and every [math]\displaystyle{ F \in \Sigma }[/math] with [math]\displaystyle{ \mu (F) \lt \infty }[/math],
- [math]\displaystyle{ \lim_{n\to\infty} \mu(\{x \in F: |f(x)-f_n(x)|\geq \varepsilon\}) = 0 }[/math].
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 [math]\displaystyle{ \in }[/math] 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, [math]\displaystyle{ \mu (X)\lt \infty }[/math] 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 [math]\displaystyle{ X = \mathbb R }[/math], μ be Lebesgue measure, and f the constant function with value zero.
- The sequence [math]\displaystyle{ f_n = \chi_{[n,\infty)} }[/math] converges to f locally in measure, but does not converge to f globally in measure.
- The sequence [math]\displaystyle{ f_n = \chi_{\left[\frac{j}{2^k},\frac{j+1}{2^k}\right]} }[/math] where [math]\displaystyle{ k = \lfloor \log_2 n\rfloor }[/math] and [math]\displaystyle{ j=n-2^k }[/math] (The first five terms of which are [math]\displaystyle{ \chi_{\left[0,1\right]},\;\chi_{\left[0,\frac12\right]},\;\chi_{\left[\frac12,1\right]},\;\chi_{\left[0,\frac14\right]},\;\chi_{\left[\frac14,\frac12\right]} }[/math]) 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 [math]\displaystyle{ f_n = n\chi_{\left[0,\frac1n\right]} }[/math] converges to f almost everywhere and globally in measure, but not in the p-norm for any [math]\displaystyle{ p \geq 1 }[/math].
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
- [math]\displaystyle{ \{\rho_F : F \in \Sigma,\ \mu (F) \lt \infty\}, }[/math]
where
- [math]\displaystyle{ \rho_F(f,g) = \int_F \min\{|f-g|,1\}\, d\mu }[/math].
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 [math]\displaystyle{ G\subset X }[/math] of finite measure and [math]\displaystyle{ \varepsilon \gt 0 }[/math] there exists F in the family such that [math]\displaystyle{ \mu(G\setminus F)\lt \varepsilon. }[/math] When [math]\displaystyle{ \mu(X) \lt \infty }[/math], we may consider only one metric [math]\displaystyle{ \rho_X }[/math], so the topology of convergence in finite measure is metrizable. If [math]\displaystyle{ \mu }[/math] is an arbitrary measure finite or not, then
- [math]\displaystyle{ d(f,g) := \inf\limits_{\delta\gt 0} \mu(\{|f-g|\geq\delta\}) + \delta }[/math]
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
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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