# Dictionary Definition

measurable adj

1 possible to be measured; "measurable depths"
[syn: mensurable]
[ant: immeasurable]

2 of distinguished importance; "a measurable
figure in literature"

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# Extensive Definition

In mathematics the concept of a
measure generalizes notions such as "length", "area", and "volume"
(but not all of its applications have to do with physical sizes).
Informally, given some base set, a "measure" is any consistent
assignment of "sizes" to (some of) the subsets of the base set.
Depending on the application, the "size" of a subset may be
interpreted as (for example) its physical size, the amount of
something that lies within the subset, or the probability that some
random process will yield a result within the subset. The main use
of measures is to define general concepts of integration over domains with
more complex structure than intervals of the real line. Such
integrals are used extensively in probability
theory, and in much of mathematical
analysis.

It is often not possible or desirable to assign a
size to all subsets of the base set, so a measure does not have to
do so. There are certain consistency conditions that govern which
combinations of subsets it is allowed for a measure to assign sizes
to; these conditions are encapsulated in the auxiliary concept of a
σ-algebra.

In differential
topology, the related concept of volume form
is used more frequently.

Measure theory is that branch of real
analysis which investigates σ-algebras,
measures, measurable
functions and integrals.

## Definition

Formally, a measure μ is a function
defined on a σ-algebra
Σ over a set X and taking values in the extended
interval [0,∞] such that the following properties are
satisfied:

- The empty set has measure zero:

- \mu(\varnothing) = 0 .

- Countable additivity or σ-additivity: if E_1, E_2, E_3,\dots\,\! is a countable sequence of pairwise disjoint sets in \Sigma, the measure of the union of all the E_i\,\!'s is equal to the sum of the measures of each E_i\,\!:

- \mu\left(\bigcup_^\infty E_i\right) = \sum_^\infty \mu(E_i).

The triple (X,Σ,μ) is
then called a measure space, and the members of Σ are
called measurable sets.

A probability measure is a measure with total
measure one (i.e., μ(X) = 1); 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. This approach is taken by Bourbaki (2004) and
a number of other authors. For more details see Radon
measure.

## Properties

Several further properties can be derived from the definition of a countably additive measure.### Monotonicity

\mu is monotonic: If E_1 and E_2 are measurable sets with E_1\subseteq E_2 then \mu(E_1) \leq \mu(E_2).### Measures of infinite unions of measurable sets

\mu is countably subadditive: If E_1, E_2, E_3,\dots\, is a countable sequence of sets in \Sigma, not necessarily disjoint, then- \mu\left( \bigcup_^\infty E_i\right) \le \sum_^\infty \mu(E_i).

\mu is continuous from below: If E_1, E_2,
E_3,\dots\, are measurable sets and E_n is a subset of E_ for all
n, then the union
of the sets E_n is measurable, and

- \mu\left(\bigcup_^\infty E_i\right) = \lim_ \mu(E_i).

### Measures of infinite intersections of measurable sets

\mu is continuous from above: If E_1, E_2, E_3,\dots are measurable sets and E_ is a subset of E_n for all n, then the intersection of the sets E_n is measurable; furthermore, if at least one of the E_n has finite measure, then- \mu\left(\bigcap_^\infty E_i\right) = \lim_ \mu(E_i).

This property is false without the assumption
that at least one of the E_n has finite measure. For instance, for
each n ∈ N, let

- E_n = [n, \infty) \subseteq \mathbb

which all have infinite 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
∞). It is called σ-finite if X can be
decomposed into a countable union of measurable sets of finite
measure. A set in a measure space has 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.

## 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).

## 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 the unique complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1.
- Circular angle measure is invariant under rotation.
- 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 which is a refinement of the Lebesgue measure to some 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 \mu_a (cf. Dirac delta function) is given by \mu_a(S) = \chi_S(a) = [a \in S], where \chi_S is the characteristic function of S and the brackets signify the Iverson notation. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other measures include: Borel
measure, Jordan
measure, Ergodic
measure, Euler
measure, Gauss
measure, Baire
measure, Radon
measure.

## Non-measurable sets

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. 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 mainly 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.

Another generalization is the finitely additive
measure. This is the same as a measure except that instead of
requiring countable additivity we require only finite additivity.
Historically, this definition was used first, but proved to be not
so useful. 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.

The remarkable result in integral
geometry known as Hadwiger's
theorem states that the space of translation-invariant,
finitely additive, not-necessarily-nonnegative set functions
defined on finite
unions of compact convex sets in
\mathbb^n consists (up to scalar multiples) of one "measure" that
is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and
linear combinations of those "measures". "Homogeneous of degree k"
means that rescaling any set by any factor c>0 multiplies the
set's "measure" by c^k. The one that is homogeneous of degree n is
the ordinary n-dimensional volume. The one that is homogeneous of
degree n − 1 is the "surface volume".
The one that is homogeneous of degree 1 is a mysterious function
called the "mean width", a misnomer. The one that is homogeneous of
degree 0 is the Euler
characteristic.

A measure is a special kind of content.

## See also

## References

- R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.
- Integration I Chapter III.
- R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.
- Real Analysis: Modern Techniques and Their Applications Second edition.
- D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
- Paul Halmos, 1950. Measure theory. Van Nostrand and Co.
- 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.
- 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.
- Some useful Cambridge Tripos Notes on Probability and Measure Theory link

measurable in Arabic: نظرية القياس

measurable in Czech: Teorie míry

measurable in Danish: Målteori

measurable in German: Maßtheorie

measurable in Spanish: Teoría de la medida

measurable in Persian: نظریه اندازه

measurable in French: Mesure
(mathématiques)

measurable in Korean: 측도

measurable in Icelandic: Mál (stærðfræði)

measurable in Italian: Misura (matematica)

measurable in Hebrew: מידה (מתמטיקה)

measurable in Hungarian: Mérték
(matematika)

measurable in Dutch: Maat (wiskunde)

measurable in Japanese: 測度論

measurable in Polish: Miara (matematyka)

measurable in Portuguese: Medida
(matemática)

measurable in Romanian: Măsură
(matematică)

measurable in Russian: Мера множества

measurable in Serbian: Мера (математика)

measurable in Finnish: Mittateoria

measurable in Swedish: Mått (matematik)

measurable in Thai: ทฤษฎีการวัด

measurable in Vietnamese: Độ đo

measurable in Ukrainian: Теорія міри

measurable in Chinese: 测度