Convergence in measure

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In mathematics, particularly measure theory, convergence in measure is a weak notion of convergence of measurable functions. It is a generalization of the concept of congervence in probability which applies to random variables.

1 Definition
2 Properties
3 Topology
4 References

[edit] Definition

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 to f in measure if for any ε > 0 and any δ > 0, there is a natural number N such that for $n \geq N$ , we have

$\mu\{x \in X: |f(x) - f_n(x)| \geq \varepsilon\} < \delta$ .

Equivalently, for any ε > 0,

$\lim_{n\to\infty} \mu\{x \in X: |f(x)-f_n(x)|\geq \varepsilon\} = 0$ .

[edit] Properties

Throughout, f and f_n (n $\in$ N) are measurable functions X → R.

If μ is σ-finite and (f_n) converges to f in measure, there is a subsequence converging to f almost everywhere.

If μ is σ-finite, (f_n) converges to f in measure if and only if every subsequence has in turn a subsequence that converges to f almost everywhere.

Fatou's lemma and the monotone convergence theorem hold if convergence almost everywhere is replaced by 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 in measure to f.

If each f_n vanishes outside some set of finite measure, and (f_n) converges to f almost everywhere, then (f_n) converges to f in measure. The converse is false.

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 in measure. The converse is false.

Suppose that for any ε > 0, there is a measurable subset E of X with μE < ε and a natural number N such that for all n,m ≥ N and x ∈ E, we have |f_n(x) - f_m(x)| < ε (we say (f_n) is Cauchy in measure). Then there is a function g such that (f_n) converges to g in measure.

[edit] Topology

There is a topology, called the topology of convergence in measure, on the collection of measurable functions from X such that convergence in measure corresponds to convergence on that topology. This topology is defined by the family of pseudometrics

$\{\rho_F : F \in \Sigma,\ \mu F < \infty\},$