Measure (mathematics) 1
Measure (mathematics)
Informally, a measure has the property of being
monotone in the sense that if A is a subset of B,
the measure of A is less than or equal to the
measure of B. Furthermore, the measure of the
empty set is required to be 0.
In mathematical analysis, a measure on a set is a systematic way to
assign a number to each suitable subset of that set, intuitively
interpreted as its size. In this sense, a measure is a generalization of the
concepts of length, area, and volume. A particularly important example
is the Lebesgue measure on a Euclidean space, which assigns the
conventional length, area, and volume of Euclidean geometry to
suitable subsets of the -dimensional Euclidean space . For
instance, the Lebesgue measure of the interval in the real
numbers is its length in the everyday sense of the word – specifically,
1.
Technically, a measure is a function that assigns a non-negative real
number or +∞ to (certain) subsets of a set (see Definition below). It
must assign 0 to the empty set and be (countably) additive: the measure
of a 'large' subset that can be decomposed into a finite (or countable)
number of 'smaller' disjoint subsets, is the sum of the measures of the
"smaller" subsets. In general, if one wants to associate a consistent size
to each subset of a given set while satisfying the other axioms of a
measure, one only finds trivial examples like the counting measure.
This problem was resolved by defining measure only on a
sub-collection of all subsets; the so-called measurable subsets, which
are required to form a -algebra. This means that countable unions,
countable intersections and complements of measurable subsets are
measurable. Non-measurable sets in a Euclidean space, on which the
Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up
with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel,
Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the
foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic
theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets
of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general
and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign
to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.
Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.
Definition
Countable additivity of a measure : The measure of a countable disjunctive
union is the same as the sum of all measures of each subset.
Let be a set and a -algebra over
. A function from to the extended
real number line is called a measure if it
satisfies the following properties:
• Non-negativity:
.
• Null empty set:
Measure (mathematics) 2
.
• Countable additivity (or -additivity): For all countable collections of pairwise disjoint sets in :
.
One may require that at least one set has finite measure. Then the null set automatically has measure zero
because of countable additivity, because , so
.
If only the second and third conditions of the definition of measure above are met, and takes on at most one of the
values , then is called a signed measure.
The pair is called a measurable space, the members of are called measurable sets. If and
are two measurable spaces, then a function is called measurable if for every -measurable
set , the inverse image is -measurable – i.e.: . The composition of measurable
functions is measurable, making the measurable spaces and measurable functions a category, with the measurable
spaces as objects and the set of measurable functions as arrows.
A triple is called a measure space. A probability measure is a measure with total measure one – i.e.
– a probability space is a measure space with a probability measure.
For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure
and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon
measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of
continuous functions with compact support.Wikipedia:Please clarify This approach is taken by Bourbaki (2004) and
a number of other sources. For more details, see the article on Radon measures.
Properties
Several further properties can be derived from the definition of a countably additive measure.
Monotonicity
A measure μ is monotonic: If E1 and E2 are measurable sets with E1 ⊆ E2 then
Measures of infinite unions of measurable sets
A measure μ is countably subadditive: For any countable sequence E1, E2, E3,… of sets En in Σ (not necessarily
disjoint):
A measure μ is continuous from below: If E1, E2, E3,… are measurable sets and En is a subset of En + 1 for all n,
then the union of the sets E
n
is measurable, and
Measure (mathematics) 3
Measures of infinite intersections of measurable sets
A measure μ is continuous from above: If E1, E2, E3, … are measurable sets and En + 1 is a subset of En for all n, then
the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then
This property is false without the assumption that at least one of the En has finite measure. For instance, for each n ∈
N, let
which all have infinite Lebesgue measure, but the intersection is empty.
Sigma-finite measures
A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are
analogous to probability measures in the sense that any finite measure is proportional to the probability measure
. A measure is called σ-finite if X can be decomposed into a countable union of measurable sets of
finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of
sets with finite measure.
For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed
intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the
entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set
of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure
contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The
σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the
Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a
measure space may have 'uncountable measure'.
Completeness
A measurable set X is called a null set if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need
not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if
every negligible set is measurable.
A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible
set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One
defines μ(Y) to equal μ(X).
Additivity
Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I
and any set of nonnegative ri, define:
That is, we define the sum of the to be the supremum of all the sums of finitely many of them.
A measure on is -additive if for any and any family , the following hold:
1.
Measure (mathematics) 4
2.
Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.
Examples
Some important measures are listed here.
• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in
R such that μ([0,1]) = 1; and every other measure with these properties extends Lebesgue measure.
• Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze
mapping.
• The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also
of counting measure and circular angle measure) and has similar uniqueness properties.
• The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in
particular, fractal sets.
• Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes
all its values in the unit interval [0,1]). Such a measure is called a probability measure. See probability axioms.
• The Dirac measure δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the characteristic function of
S. The measure of a set is 1 if it contains the point a and 0 otherwise.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler
measure, Gaussian measure, Baire measure, Radon measure and Young measure.
In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another
non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to
signed measures, see "generalizations" below.
Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical
and Hamiltonian mechanics.
Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.
Non-measurable sets
If the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examples
of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the
Banach–Tarski paradox.
Generalizations
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or
infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed
measure, while such a function with values in the complex numbers is called a complex measure. Measures that take
values in Banach spaces have been studied extensively.[citation needed] A measure that takes values in the set of
self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis
for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from
generalizations, the term positive measure is used. Positive measures are closed under conical combination but not
general linear combination, while signed measures are the linear closure of positive measures.
Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a
measure except that instead of requiring countable additivity we require only finite additivity. Historically, this
Measure (mathematics) 5
definition was used first. It turns out that in general, finitely additive measures are connected with notions such as
Banach limits, the dual of L∞ and the Stone–Čech compactification. All these are linked in one way or another to the
axiom of choice.
A charge is a generalization in both directions: it is a finitely additive, signed measure.
References
• Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.
• Bauer, H. (2001), Measure and Integration Theory, Berlin: de Gruyter, ISBN 978-3110167191
• Bear, H.S. (2001), A Primer of Lebesgue Integration, San Diego: Academic Press, ISBN 978-0120839711
• Bogachev, V. I. (2006), Measure theory, Berlin: Springer, ISBN 978-3540345138
• Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III.
• R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.
• Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons,
ISBN 0471317160 Second edition.
• D. H. Fremlin, 2000. Measure Theory [1]. Torres Fremlin.
• Paul Halmos, 1950. Measure theory. Van Nostrand and Co.
• Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag,
ISBN 3-540-44085-2
• R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of
Economics, v. 3, pp. 428–32.
• M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
• K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983), Theory of Charges: A Study of Finitely Additive Measures,
London: Academic Press, pp. x + 315, ISBN 0-12-095780-9
• Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A.
Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.
• Teschl, Gerald, Topics in Real and Functional Analysis [2], (lecture notes)
External links
• Hazewinkel, Michiel, ed. (2001), "Measure" [3], Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4
• Tutorial: Measure Theory for Dummies [4]
References
[1] http:/ / www. essex. ac. uk/ maths/ people/ fremlin/ mt. htm
[2] http:/ / www. mat. univie. ac. at/ ~gerald/ ftp/ book-fa/ index. html
[3] http:/ / www. encyclopediaofmath. org/ index. php?title=p/ m063240
[4] http:/ / www. ee. washington. edu/ techsite/ papers/ documents/ UWEETR-2006-0008. pdf
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Measure (mathematics)
Definition
Properties
Monotonicity
Measures of infinite unions of measurable sets
Measures of infinite intersections of measurable sets
Sigma-finite measures
Completeness
Additivity
Examples
Non-measurable sets
Generalizations
References
External links
License