Convergence in measure

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) \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 f_n (n\in\N) are measurable functions X\to\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) or, more generally, if f and all the f_n vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears. • If \mu 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 \mu is \sigma-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 \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]\subseteq\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 are in Lp(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_n g_n 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 f_n(x) converge to zero. Hence (f_n) 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) 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) When \mu(X) , 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. ==See also==