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
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 ...
The sequence
is said to to
if for every
and to to
if for every
and every
with
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,
and
(
) are measurable functions
.
* 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,
or, more generally, if
and all the
vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
* If
is
''σ''-finite and (''f''
''n'') converges (locally or globally) to
in measure, there is a subsequence converging to
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
is
-finite,
converges to
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
almost everywhere.
* In particular, if
converges to
almost everywhere, then
converges to
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
is
-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
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
of step functions and
of continuous functions converging globally in measure to
.
* If
and
are in
''L''''p''(''μ'') for some
and
converges to
in the
-norm, then
converges to
globally in measure. The converse is false.
* If
converges to
in measure and
converges to
in measure then
converges to
in measure. Additionally, if the measure space is finite,
also converges to
.
Counterexamples
Let
,
be Lebesgue measure, and
the constant function with value zero.
* The sequence
converges to
locally in measure, but does not converge to
globally in measure.
* The sequence
::
:where
and
, the first five terms of which are
::
:converges to
globally in measure; but for no
does
converge to zero. Hence
fails to converge to
almost everywhere.
* The sequence
::
:converges to
almost everywhere and globally in measure, but not in the
-norm for any
.
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
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
of finite measure and
there exists ''F'' in the family such that
When
, we may consider only one metric
, so the topology of convergence in finite measure is metrizable. If
is an arbitrary measure finite or not, then
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