Measure

$\newcommand{\E}{\mathcal E}$

measure of a set

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 $X$ be a set and let $\E$ be a class of subsets of $X$. A non-negative (not necessarily finite) set function $\lambda$ defined on $\varepsilon$ is called additive, finitely additive or countably additive if

\[\lambda \left( \bigcup\limits_{i = 1}^n E_i \right) = \sum\limits_{i = 1}^n {\lambda ({E_i})} \]

whenever

\[ E_i \in \varepsilon, \quad \bigcup\limits_{i=1}^n E_i\in \varepsilon, \quad E_i \bigcap E_j=\emptyset, i \ne j, \]

for, respectively, $n=2$, $n$ arbitrary finite, and $n \le \infty$.

A collection $\mathcal{P}$ of subsets of $X$ is called a semi-ring of sets if

1. $ \emptyset \in \mathcal{P} $
2. $ E_1, E_2 \in \mathcal{P} \implies E_1 \cap E_2 \in \mathcal{P} $
3. $ E, E_1 \in \mathcal{P}, E_1 \subset E \implies E$ is representable as $ E = \bigcup_{i=1}^n E_i, E_i \cap E_j = \emptyset $ for $i \ne j, E_i \in \mathcal{P}, i = 1 \dots n, n < \infty$ .

A collection $ \mathcal{R}$ of subsets of $X$ is called a ring of sets if

1. $\emptyset \in \mathcal{R}$
2. $E_1, E_2 \in \mathcal{R} \implies E_1 \cup E_2 \in \mathcal{R}, E_1 \setminus E_2 \in \mathcal{R}$.

An example of a semi-ring is: $X = \mathbf{R}^k $, $\mathcal{P}$ is the collection of all intervals of the form

\[ \{x = (x_1, \dots, x_k) \in \mathbf{R}^k \mid a_i \le x_i < b_i, i = 1, \dots, k\} \]

where $ a_i, b_i \in \mathbf{R} $ for $ i = 1, \dots, k $. The collection of all possible finite unions of such intervals is a ring.

A collection $\mathcal{S}$ of subsets of $X$ is called a $\sigma$-ring if

1. $\emptyset \in \mathcal{S} $
2. $E_1, E_2 \in \mathcal{S} \implies E_1 \setminus E_2 \in \mathcal{S}$
3. $E_i \in \mathcal{S} \quad (i = 1, 2, \dots) \implies \bigcup_{i=1}^\infty E_i \in S$.

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.

References

[1]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[3]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[4]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[5]	J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French) MR0198505
[6]	B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[7]	V.S. Varadarajan, "Measures on topological spaces" Mat. Sb. , 55 : 1 (1961) pp. 35–100 (In Russian) MR0148838
[8]	K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967) MR0226684 Zbl 0153.19101
[9]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[10]	R. Sikorski, "Boolean algebras" , Springer (1969) MR0249336 MR0242724 Zbl 0191.31505
[11]	D.A. Vladimirov, "Boolesche Algebren" , Akademie Verlag (1978) (Translated from Russian) MR0524392 Zbl 0385.06018
[12]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[13]	J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977) MR0453964 Zbl 0369.46039

Comments

Properties 1) and 2) listed under the heading "Properties of measure spaces" are usually called Fatou's lemma, cf. Fatou theorem.

The procedure for extending a measure, as described under the heading "Extension of measures" , is due to C. Carathéodory, and one often speaks of Carathéodory extension, with the accompanying phrases Carathéodory extension theorem and Carathéodory outer (inner) measure (cf. Carathéodory measure).

Recall that a ring (respectively, a -ring) of subsets of a set such that implies , is called a Boolean algebra or an algebra (respectively, a -algebra or a -field, cf. also Algebra of sets). Usually, in a measure space the -ring can be proved to be a -field (this holds, in particular, if ).

The phrase "totally (s-) finite" is seldom used.

Borel has given very nice ideas in order to construct the measure , but Lebesgue was the first to give a satisfactory construction of it, as a byproduct of the construction of .

A product space is also often written as a (kind of) tensor product: .

A family of measurable spaces with compatible probability measures on each finite product is called a projective system of measure spaces, and the corresponding probability measure on , if it exists, is called the projective limit; it exists if is countable (the Ionescu–Tulcea theorem, cf. [5]).

Suppose that is a topological space and is its Borel -field; then is perfect for every finite measure if is a Polish space or, more generally, a Luzin space (in which case is often called a standard measurable space) or, still more generally, a Suslin space (in which case is sometimes called a Blackwell measurable space) (cf. (the editorial comments to) Descriptive set theory).

The converse part of Prokhorov's theorem is not true when is the space of rational numbers, or, more generally, when is a Luzin space which is not Polish. See [a1].

In the abstract setting, whenever is a sequence of finite measures on , where is a -field, such that

exists for any , then is also a measure (the Vitali–Hahn–Saks theorem, cf. [3] or [5]).

References

[a1]	D. Preiss, "Metric spaces in which Prokhorov's theorem is not valid" Z. Wahrscheinlichkeitstheor. Verw. Gebiete , 27 (1973) pp. 109–116
[a2]	D. Cohn, "Measure theory" , Birkhäuser (1980) MR0578344 Zbl 0436.28001
[a3]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , I , Springer (1979) MR0551496 Zbl 0416.43001
[a4]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202