A notion that generalizes those of the length of
segments, the area of figures and the volume of bodies, and that
corresponds intuitively to the mass of a set for some mass distribution
throughout the space. The notion of the measure of a set arose in
the theory of functions of a real variable in connection
with the study and improvement of the notion of an
integral.
Definition and general properties.
Let
be a set and let
be a class of subsets of
.
A non-negative (not necessarily finite) set function
defined on
is called
additive,
finitely additive
or
countably additive
if
whenever
for, respectively,
,
arbitrary finite, and
.
A collection
of subsets of
is called a
semi-ring of sets
if
1)
;
2)
imply
;
3)
,
imply that
is representable as
,
for
,
,
,
.
A collection
of subsets of
is called a
ring of sets
if
1)
;
2)
imply
,
.
An example of a semi-ring is:
,
is the collection of all intervals of the form
where
for
.
The collection of all possible finite unions of such intervals is a ring.
A collection
of subsets of
is called a
-ring
if
1)
;
2)
imply
;
3)
,
implies
.
Every
-ring
is a ring; every ring is a semi-ring.
A
finitely-additive measure
is a non-negative finitely-additive set function
such that
.
The domain of definition
of a finitely-additive measure may be a semi-ring, a ring or a
-ring.
In the definition of a finitely-additive measure on a ring or on a
-ring
the condition of finite additivity can be weakened
to additivity, which leads to the same notion.
If
is a finitely-additive measure, if the sets
belong to its domain of definition, and if
,
then
Let
be a finitely-additive measure with domain
.
A finitely-additive measure
with domain
is called an
extension
of
if
and
for all
.
Every finitely-additive measure
defined on a semi-ring
admits a unique extension to a finitely-additive measure
on the smallest ring
containing
.
This extension is defined as follows: Every
is representable as
,
,
,
,
and one sets
A finitely-additive measure that has the property of countable additivity is called a
measure.
Examples of measures: Let
be an arbitrary non-empty set, let
be a
-ring,
a ring or a semi-ring of subsets of
,
let
be a countable subset of
,
and let
be non-negative numbers. Then the function
where
if
and
if
,
is a measure defined on
.
The measures
are called
elementary,
degenerate
or
Dirac measures
(sometimes,
Dirac masses).
Not every finitely-additive measure is a measure. For example, if
is the set of rational points of the segment
,
is the semi-ring of all possible intersections of subintervals of
with
,
and for every
,
,
then
is finitely additive, but not countably additive on
.
A (finitely-additive) measure
with domain
is said to be
finite
(respectively,
-finite)
if
for all
(respectively, if for every
there is a sequence of sets
in
such that
and
,
).
A (finitely-additive) measure
is said to be
totally finite
(totally
-finite)
if it is finite (respectively,
-finite)
and
.
A pair
,
where
is a set and
is a
-ring
of subsets of
such that
,
is called a
measurable space.
A triple
,
where
is a measurable space and
is a measure on
,
is called a
measure space.
A space with a totally-finite measure
normalized by the condition
is called a
probability space.
In abstract measure theory, where the basic notions are a measurable space
or a measure space
,
the elements of
are also referred to as
measurable sets
(cf. also
Measurable set).
Properties of measure spaces.
Let
be an arbitrary sequence of measurable sets. Then
1)
;
2)
if
for some
,
then
3)
if
exists and the condition in 2) is satisfied, then
A finitely-additive measure
defined on a ring
is a measure if and only if
for every monotone increasing sequence
of elements of
such that
.
Let
be a measure space, let
be a measurable space and let
be a
measurable mapping
from
into
,
i.e.
for all
.
The
measure generated by the mapping
(denoted here by
)
is the measure on
defined by
Let
be a measure space and let
.
Define
on the sets
from the
-ring
by
Then
is a measure space;
is called the
restriction of the measure
to
.
An
atom
of the space
(or of the measure
)
is any set
of positive measure such that if
,
,
then either
or
.
A measure space without atoms is called
non-atomic
or
continuous
(in this case
is also called
non-atomic
or
continuous).
If
is a space with a non-atomic
-finite
measure and
,
then for every
with
(possibly
)
there is an element
such that
and
.
A measure space
(or the measure
)
is said to be
complete
if
,
,
imply
.
Every measure space
can be completed by adjoining to
all the sets of the form
with
,
,
,
,
and putting for such sets
.
The class of sets of the indicated form is a
-ring,
and
is a complete measure on it. The sets of null measure are called
null sets.
If the set of points of
at which a property
is not satisfied is a null set, then property
is said to hold
almost-everywhere.
Extension of measures.
A measure
is an extension of a measure
if
is an extension of
in the class of finitely-additive measures (see
above). Every measure defined on a semi-ring
admits a unique extension to a measure on the ring
generated by
(the extension is realized in the same way as
in the case of finitely-additive measures). Further, every measure
defined on a ring
can be extended to a measure
on the
-ring
generated by
;
if
is
-finite,
then
is unique and
-finite.
The value of
on any set
can be given by the formula
A class of subsets of
is called
hereditary
if it contains, together with any set in the class, all its subsets. An
outer measure
is a set function
,
defined on a
hereditary
-ring
(i.e. a class of sets which is simultaneously hereditary and a
-ring),
which has the following properties:
1)
,
;
2)
implies
;
3)
.
Given a measure
on the ring
one can construct an outer measure
on the hereditary
-ring
generated by
(
consists of all sets that can be covered by a countable union of elements of
)
by means of the formula
The outer measure
is called the
outer measure induced by the measure
.
Let
be an outer measure on a hereditary
-ring
of subsets of
.
A set
is called
-measurable
if
for every
.
The collection
of
-measurable
sets is a
-ring
which contains all sets of null outer measure. The set function
on
defined by the equality
is a complete measure and is called the
measure induced by the outer measure
.
Suppose that
is a measure on a ring
and that
is the outer measure on
induced by
.
Let
and
denote the collection of
-measurable
sets and the measure on
induced by
,
respectively. Then
is an extension of
,
and since
it follows that the function
on
given by formula
(*)
is also a measure extending
.
If the original measure
on
is
-finite,
then the space
is the completion of the space
(see
(*)). If
is given on the
-ring
,
then the induced outer measure
on the hereditary
-ring
generated by
is given by the formula
Alongside with the outer measure
,
one defines the
inner measure induced by the measure
on
.
It is defined as
For every set
a
measurable kernel
and a
measurable envelope
are defined as elements of
such that
and
for all
such that
,
.
A measurable kernel exists always, while a measurable envelope exists whenever
has
-finite
outer measure; moreover,
and
.
Let
be a measure on a ring
and let
be its extension to the
-ring
generated by
.
The inner measure
on the subsets
of finite
-measure
can be expressed in terms of the outer measure
(and hence
):
Furthermore, a set
belonging to the hereditary
-ring
with finite outer
-measure
is
-measurable
if and only if
.
In case the original measure
on
is totally finite, one has the following necessary and sufficient condition for the
-measurability
of a set
:
For totally-finite measures on
this condition is frequently taken as the definition of
-measurability
of the set
.
If
is a space with a
-finite
measure and
is a finite collection of elements of the hereditary
-ring
generated by
,
then on the
-ring
generated by
and the sets
one can define a measure
which agrees with
on
.
Jordan, Lebesgue and Lebesgue–Stieltjes measures.
An example of an extension of a measure is provided by the Lebesgue measure in
.
The intervals of the form
form a semi-ring
in
.
For each such interval, let
(
coincides with the volume of
).
The function
is
-finite
and countably additive on
and admits a unique extension to a measure
on the
-ring
generated by
;
is identical with the
-ring
of Borel sets (cf.
Borel set)
(or Borel-measurable sets) in
.
The measure
was first defined by
E. Borel
in
1898
(see
Borel measure).
The completion
of
(defined on
)
is called the
Lebesgue measure,
and was introduced by
H. Lebesgue
in
1902
(see
Lebesgue measure).
A set belonging to the domain
of
is called
Lebesgue measurable.
A bounded set
belongs to
if and only if
,
where
is some interval containing
;
in this case
.
A set
belongs to
if and only if for some sequence
,
,
such that
,
one has
for all
,
where
.
The cardinality of the family of all Borel sets in
is
(the cardinality of the continuum), whereas the cardinality
of the family of all Lebesgue-measurable sets is
,
so that the inclusion
is strict, i.e. there exist Lebesgue-measurable sets that are not Borel measurable.
The Lebesgue measure
is invariant under linear orthogonal transformations
of
as well as under translations by elements
,
i.e.
for all
.
Using the
axiom of choice
one can show that there exist sets which are
not Lebesgue measurable. On the straight line, for example, such a set
can be obtained by picking one point in each coset in
of the additive subgroup of rational numbers
(Vitali's example).
Historically the Borel and Lebesgue measures in
were preceded by the measure defined by
C. Jordan
in
1892
(see
Jordan measure).
The idea of the definition of the Jordan measure is very
close to that of the classic definition of area and
volume, which goes back to ancient Greece. Thus, a set
is called
Jordan measurable
if there exist two sets, representable as finite
unions of disjoint rectangles, one contained in
and the other containing
,
such that the difference of their volumes (defined
in an obvious manner) is arbitrarily small. The
Jordan measure
of such a set is the infimum of the volumes of finite unions of rectangles covering
.
A Jordan-measurable set is also Lebesgue measurable, and its Jordan
and Lebesgue measures are equal. The domain of the
Jordan measure is merely a ring, and not a
-ring,
which restricts considerably its domain of applicability.
The Lebesgue measure is a particular case of the
more general Lebesgue–Stieltjes measure. The latter is
defined by means of a real-valued function
on
with the properties:
1)
;
2)
for
,
,
where
is the difference operator with step
taken at the point
with respect to the
-th
coordinate;
3)
as
,
.
Given such a function
,
the measure
of the interval
is defined by the formula
It turns out that
is countably additive on the semi-ring of all such
intervals and that it admits an extension to the
-algebra
of Borel sets; the completion of this extension yields what is called the
Lebesgue–Stieltjes measure
corresponding to
.
For the particular choice
one obtains the Lebesgue measure.
Measures in product spaces.
By definition, the
product of two measurable spaces
,
is the measurable space consisting of the set
(the product of
and
)
and the
-ring
of subsets of
(the product of the
-rings
and
)
generated by the semi-ring
of sets of the form
where
.
If
and
are measure spaces, the formula
defines a measure on
;
if
and
are
-finite,
extends uniquely to a measure on
,
denoted by
.
The measure
and the space
are called, respectively, the
product of the measures
and
,
and the
product of the measure spaces
and
.
The completion of the product of the Lebesgue measure in
and the Lebesgue measure in
is the Lebesgue measure in
.
Analogously one defines the product of an arbitrary finite number of measure spaces.
Let
,
,
be an arbitrary family of measure spaces such that
,
.
The
product space
is, by definition, the set of all functions on
such that the value at each
is an element
.
A
measurable rectangle
in
is any set of the form
,
where
and only finitely many sets
are different from
.
The family of measurable rectangles forms a semi-ring
.
The
-ring
generated by
is denoted by
and is called the
product of the
-rings
.
Now, let
be the function on
defined by
for
.
The function
thus defined is a measure which admits a unique extension to a measure on
,
denoted by
.
The measure space
is called the
product of the spaces
,
.
The product of an arbitrary number of measure spaces is a
particular case of the following, more general, scheme, which
plays an important role in probability theory. Let
,
,
be a family of measurable spaces (each
is a
-algebra),
and suppose that for each finite subset
there is given a probability measure
on the measurable spaces
(the product of measures corresponds to the case that
for all finite
).
Suppose further that each two measures
are
compatible
in the sense that if
and
is the projection of
onto
,
then
for all
(by definition,
is the mapping of
onto
such that
for all
).
The following question arises: Is there a probability measure on
such that
for every finite
and every
,
where
denotes the projection of
onto
?
It turns out that such a measure does not
always exist, and additional conditions must be imposed to guarantee
its existence. One such condition is perfectness of the measures
(corresponding to the one-point sets
).
The notion of a
perfect measure
was first introduced by
B.V. Gnedenko
and
A.N. Kolmogorov
[6].
A space
with a totally-finite measure, as well as the measure
itself, is called
perfect
if for every
-measurable
real-valued function
on
there is a Borel set
such that
.
The perfectness assumption eliminates a series of
"pathological"
phenomena that arise in general measure theory.
Measures in topological spaces.
The study of measures in topological spaces is usually concerned with
measures defined on sets connected in some way or another with
the topology of the underlying space. One of the typical approaches is the following. Let
be an arbitrary topological space and let
be the class of subsets of the form
,
where
is a continuous real-valued function on
and
is a closed set. Let
be the algebra generated by the class
and let
be the
-algebra
generated by
(
is called the
-algebra of Baire sets,
cf. also
Algebra of sets).
Now let
be the class of totally-finite finitely-additive measures
on
that are regular in the sense that
for all
.
In
one distinguishes the subclasses
,
and
formed by the (finitely-additive) measures possessing
additional smoothness properties. By definition,
if
for every sequence
,
(this property is equivalent to the countable additivity of
;
the measures from
admit unique extensions to
and hereafter it is assumed that they are given on
);
if
for every net
,
;
and
if for every
there is a compact set
such that
whenever
,
.
The inclusions
hold. The elements of
are called
Baire measures.
There is an intimate connection between the measures belonging to
and the linear functionals on the space
of bounded continuous functions on
.
Namely, the formula
establishes a one-to-one correspondence between the finitely-additive measures
and the non-negative linear functionals
on
(non-negative means that
whenever
,
).
Moreover, for every set
,
where
is the indicator function of
.
This correspondence takes the measures from
into
-smooth functionals
(i.e. functionals
with the property that
if
in
),
the measures from
into
-smooth functionals
(i.e. functionals such that
for every net
in
),
and the measures from
into
dense functionals
(i.e. with the property that
for every net
in
such that
for all
and
uniformly on compact subsets; here
is the uniform norm).
The space
is usually endowed with the weak topology
,
in which a basis of neighbourhoods consists of the sets of the form
With the topology
,
is a completely-regular Hausdorff space. Convergence in the topology
is usually denoted by the symbol
.
For the convergence of a net
to
:
,
it is necessary and sufficient that
and
for all
.
Another necessary and sufficient condition for the convergence
is that
for all
such that there are
with
,
,
and
.
If the space
is completely regular and Hausdorff, then
is metrizable if and only if
is metrizable. If
is metrizable, then
admits a metric in which it is separable if and only if
is separable, and it admits a metric in which it is complete if and only if
has a complete metric. If
is metrizable, then
is metrizable if and only if it is metrizable by the
Lévy–Prokhorov metric.
The space
is sequentially closed in
(Aleksandrov's theorem).
A set
is called
tight
if
and if for every
there is a compact set
such that
for all
,
and
.
If
is tight, then
is relatively compact in
;
conversely, if
is metrizable and topologically complete, then
is relatively compact, and if every measure in
is concentrated on some separable subset of
,
then
is tight
(Prokhorov's theorem).
Under certain conditions the elements of
can be extended to Borel measures, i.e. measures defined on the
-algebra
of Borel sets (see
Borel set;
Borel measure).
For example, if
is a normal countably-paracompact Hausdorff space, then every measure
admits a unique extension to a regular Borel measure. If
is completely regular and Hausdorff, then every
-smooth
(tight) Baire measure admits a unique extension to a
-smooth
(tight) Borel measure.
The
support
of a Baire (Borel) measure is the smallest set
(respectively, the smallest closed set) the measure of which is
equal to the measure of the whole space. Every
-smooth
measure has a support.
Often, when measures in topological spaces (especially in locally compact
Hausdorff spaces) are considered, it is assumed that the Borel and
Baire measures are given on less-wide classes of sets, more precisely — on
-rings
generated by compact sets and, respectively, compact
-sets.
Let
be a locally compact Hausdorff topological group. A
left Haar measure
on
is a measure defined on the
-ring
generated by all compact subsets that does not vanish identically and is such that
for all
and
in the domain of
.
A
right Haar measure
is defined in the same manner but with the condition
replaced by
.
On any group of the type considered a left Haar measure exists
and is unique (up to a multiplicative positive constant). Every
left Haar measure is regular in the sense that
,
where
are compact sets. The right Haar measure
has analogous properties. The Lebesgue measure on
is a particular case of the
Haar measure.
See also
Measure in a topological vector space.
Isomorphism of measure spaces.
Let
be a measure space. Call two sets
-equal
(written
)
if
(where
denotes the symmetric difference of
and
,
cf.
Symmetric difference of sets).
Denote by
the class of sets
with this equality relation. In
the set-theoretic operations, performed a finite (or countable) number
of times are correctly defined: for example, if
and
,
then
.
The measure
is carried over, in an obvious manner, to
.
Let
be the subset of
consisting of the sets of finite measure. The function
on
is a metric. The measure space
is said to be
separable
if the space
with metric
is separable. If
is a space with a
-finite
measure and the
-ring
is countably generated (i.e. there is a countable family
such that
is the smallest
-ring
that contains this family), then the metric space
is separable.
Two measure spaces,
and
are said to be
isomorphic
if there is a one-to-one mapping
of
onto
such that
and
Now, let
be an arbitrary space with a totally-finite measure. There is a partition of
into disjoint sets
,
such that the restriction of
to
is isomorphic either to a measure concentrated at one point or to
a measure which is equal, up to a positive factor, to the direct product
,
where
,
,
and the set
may have arbitrary cardinality (the
Maharan–Kolmogorov theorem).
If
is separable, non-atomic and
,
then it is isomorphic to the space
with
countable, which in turn is isomorphic to the unit interval with the Lebesgue measure.
Side by side with the theory of measures regarded as functions on
subsets of some set, the theory of measures as functions
on the elements of a Boolean ring (or on a
Boolean algebra)
has been developed; these theories are in
many respects parallel. Another widespread construction of measures goes
back to
W. Young
and
P. Daniell
(see
[12]).
Theories dealing with measures with real or complex values, or
with values belonging to some algebraic structure, were developed
in addition to the theory of positive measures.