Convergence in measure

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Convergence in measure can refer to two distinct mathematical concepts which both generalize the concept of convergence in probability.

Definitions[edit]

Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be measurable functions on a measure space (X,Σ,μ). The sequence (fn) 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[edit]

Throughout, f and fn (n \in N) are measurable functions XR.

  • 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 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.
  • 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. [clarification needed]
  • If f and fn (nN) 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[edit]

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,\frac12\right]},\;\chi_{\left[\frac12,1\right]},\;\chi_{\left[0,\frac14\right]},\;\chi_{\left[\frac14,\frac12\right]}) converges to 0 locally 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\chi_{\left[0,\frac1n\right]} converges to f almost everywhere (hence also locally in measure), but not in the p-norm for any p \geq 1.

Topology[edit]

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.

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[edit]

  • 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.