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Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of
convergence in probability In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
.


Definitions

Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(X, \Sigma, \mu). The sequence f_n is said to to f if for every \varepsilon > 0, \lim_ \mu(\) = 0, and to to f if for every \varepsilon>0 and every F \in \Sigma with \mu (F) < \infty, \lim_ \mu(\) = 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)<\infty 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 (''f''''n'') converges (locally or globally) to f in measure, there is a subsequence converging to f
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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 In mathematics, Fatou's lemma establishes an inequality (mathematics), inequality relating the Lebesgue integral of the limit superior and limit inferior, limit inferior of a sequence of function (mathematics), functions to the limit inferior of ...
and the
monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its ...
hold if almost everywhere convergence is replaced by (local or global) convergence in measure. * If \mu is \sigma-finite, Lebesgue's
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
also holds if almost everywhere convergence is replaced by (local or global) convergence in measure. * If X= ,bsubseteq\R and ''μ'' is
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
, 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 ''L''''p''(''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_ converges to f locally in measure, but does not converge to f globally in measure. * The sequence ::f_n = \chi_, :where k = \lfloor \log_2 n\rfloor and j=n-2^k, the first five terms of which are ::\chi_, \;\chi_,\;\chi_,\;\chi_,\;\chi_, :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_ :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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, 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 \, where \rho_F(f,g) = \int_F \min\\, 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_ \mu(\) + \delta still defines a metric that generates the global convergence in measure.Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007 Because this topology is generated by a family of pseudometrics, it is
uniformizable In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that Uniform space#Topology of uniform spaces, induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomo ...
. Working with uniform structures instead of topologies allows us to formulate uniform properties such as Cauchyness.


See also

*
Convergence space In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of ''X'' with the Family of sets, family of Filter (set theory), filters on ...


References

* 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. {{Lp spaces Measure theory Measure, Convergence in Lp spaces