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 $$(X, \Sigma, \mu).$$ The sequence $$f_n$$ is said to  to $$f$$ if for every $$\varepsilon > 0,$$ $$\lim_{n\to\infty} \mu(\{x \in X: |f(x)-f_n(x)|\geq \varepsilon\}) = 0,$$ and to  to $$f$$ if for every $$\varepsilon>0$$ and every $$F \in \Sigma$$ with $$\mu (F) < \infty,$$ $$\lim_{n\to\infty} \mu(\{x \in F: |f(x)-f_n(x)|\geq \varepsilon\}) = 0.$$

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 $$\in$$ N) are measurable functions X &rarr; 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 f and all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
 * If &mu; is &sigma;-finite and (fn) converges (locally or globally) to f in measure, there is a subsequence converging to f almost everywhere. The assumption of &sigma;-finiteness is not necessary in the case of global convergence in measure.
 * If &mu; is &sigma;-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 &mu; is &sigma;-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 &mu; 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(&mu;) 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 $$X = \Reals$$ &mu; 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 $$f_n = \chi_{\left[\frac{j}{2^k},\frac{j+1}{2^k}\right]}$$ 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 globally 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 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 $$\{\rho_F : F \in \Sigma,\ \mu (F) < \infty\},$$ 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.

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.