Measure (mathematics) explained
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.
The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle.[1] [2] But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.
Definition
Let
be a set and
a
-algebra over
A
set function
from
to the
extended real number line is called a
measure if the following conditions hold:
of pairwise
disjoint sets in Σ,
If at least one set
has finite measure, then the requirement
is met automatically due to countable additivity:
and therefore
If the condition of non-negativity is dropped, and
takes on at most one of the values of
then
is called a
signed measure.
The pair
is called a
measurable space, and the members of
are called
measurable sets.
is called a
measure space. A
probability measure is a measure with total measure one – that is,
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 topological vector space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.
Instances
See main article: category. Some important measures are listed here.
= number of elements in
is a
complete translation-invariant measure on a
σ-algebra containing the
intervals in
such that
; and every other measure with these properties extends the Lebesgue measure.
has a canonical measure
that in local coordinates
looks like
where
is the usual Lebesgue measure.
The measure of a set is 1 if it contains the point
and 0 otherwise.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure. In physics an example of a measure is spatial distribution of mass (see for example, 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.
Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet.[3]
Basic properties
Let
be a measure.
Monotonicity
If
and
are measurable sets with
then
Measure of countable unions and intersections
Countable subadditivity
of (not necessarily disjoint) measurable sets
in
Continuity from below
If
are measurable sets that are increasing (meaning that
E1\subseteqE2\subseteqE3\subseteq\ldots
) then the
union of the sets
is measurable and
Continuity from above
If
are measurable sets that are decreasing (meaning that
E1\supseteqE2\supseteqE3\supseteq\ldots
) then the
intersection of the sets
is measurable; furthermore, if at least one of the
has finite measure then
This property is false without the assumption that at least one of the
has finite measure. For instance, for each
let
which all have infinite Lebesgue measure, but the intersection is empty.
Other properties
Completeness
See main article: Complete measure.
A measurable set
is called a
null set if
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
which differ by a negligible set from a measurable set
that is, such that the
symmetric difference of
and
is contained in a null set. One defines
to equal
"Dropping the Edge"
If
is
(\Sigma,{\calB}([0,+infty]))
-measurable, then
for
almost all
This property is used in connection with
Lebesgue integral.
Additivity
Measures are required to be countably additive. However, the condition can be strengthened as follows.For any set
and any set of nonnegative
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 of disjoint sets
the following hold:
The second condition is equivalent to the statement that the
ideal of null sets is
-complete.
Sigma-finite measures
See main article: Sigma-finite measure.
A measure space
is called finite if
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
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
for all
integers
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'.
Strictly localizable measures
See main article: Decomposable measure.
Semifinite measures
Let
be a set, let
be a sigma-algebra on
and let
be a measure on
We say
is
semifinite to mean that for all
Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)
Basic examples
- Every sigma-finite measure is semifinite.
- Assume
let
and assume
for all
is sigma-finite if and only if
for all
and
is countable. We have that
is semifinite if and only if
for all
above (so that
is counting measure on
), we see that counting measure on
is
- sigma-finite if and only if
is countable; and
- semifinite (without regard to whether
is countable). (Thus, counting measure, on the power set
of an arbitrary uncountable set
gives an example of a semifinite measure that is not sigma-finite.)
be a complete, separable metric on
let
be the Borel sigma-algebra induced by
and let
Then the
Hausdorff measure
is semifinite.
be a complete, separable metric on
let
be the Borel sigma-algebra induced by
and let
Then the packing measure
is semifinite.
Involved example
The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to
It can be shown there is a greatest measure with these two properties:We say the
semifinite part of
to mean the semifinite measure
defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:
\musf=(\sup\{\mu(B):B\in{\cal
)\})A\in{\cal
}.
\musf=(\sup\{\mu(A\cap
)\})A\in{\cal
}\}.
\musf=\mu|
\cup\{A\in{\calA}:\sup\{\mu(B):B\in{\calP}(A)\}=+infty\} x \{+infty\}\cup\{A\in{\calA}:\sup\{\mu(B):B\in{\calP}(A)\}<+infty\} x \{0\}.
Since
is semifinite, it follows that if
then
is semifinite. It is also evident that if
is semifinite then
Non-examples
Every
measure
that is not the zero measure is not semifinite. (Here, we say
measure
to mean a measure whose range lies in
: (\forallA\in{\calA})(\mu(A)\in\{0,+infty\}).
) Below we give examples of
measures that are not zero measures.
be nonempty, let
be a
-algebra on
let
be not the zero function, and let
\mu=(\sumx\inf(x))A\in{\cal
}. It can be shown that
is a measure.
\mu=\{(\emptyset,0)\}\cup({\calA}\setminus\{\emptyset\}) x \{+infty\}.
\mu=\{(\emptyset,0),(X,+infty)\}.
be uncountable, let
be a
-algebra on
let
{\calC}=\{A\in{\calA}:Aiscountable\}
be the countable elements of
and let
\mu={\calC} x \{0\}\cup({\calA}\setminus{\calC}) x \{+infty\}.
It can be shown that
is a measure.
Involved non-example
We say the
part
of
to mean the measure
defined in the above theorem. Here is an explicit formula for
: \mu0-infty=(\sup\{\mu(B)-\musf(B):B\in{\cal
)\})A\in{\cal
}.Results regarding semifinite measures
be
or
and let
*:g\mapstoTg=\left(\int
.
Then
is semifinite if and only if
is injective. (This result has import in the study of the dual space of
.)
be
or
and let
be the topology of convergence in measure on
Then
is semifinite if and only if
is Hausdorff.
be a set, let
be a sigma-algebra on
let
be a measure on
let
be a set, let
be a sigma-algebra on
and let
be a measure on
If
are both not a
measure, then both
and
are semifinite if and only if
for all
and
(Here,
is the measure defined in Theorem 39.1 in Berberian '65.)
Localizable measures
Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.
Let
be a set, let
be a sigma-algebra on
and let
be a measure on
be
or
and let
T:
\to
*:g\mapstoTg=\left(\int
.
Then
is localizable if and only if
is bijective (if and only if
"is"
).
s-finite measures
See main article: s-finite measure.
A measure is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
Non-measurable sets
See main article: Non-measurable set.
If the axiom of choice is assumed to be true, it can be proved that 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. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure.
Measures that take values in Banach spaces have been studied extensively.[4] 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, also known as a content. 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. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of
and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures.
A charge is a generalization in both directions: it is a finitely additive, signed measure.[5] (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R.)
See also
Bibliography
- Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.
- Book: Berberian . Sterling K . Measure and Integration . 1965 . MacMillan.
- Chapter III.
- Book: Dudley, Richard M.. Richard M. Dudley
. Richard M. Dudley. Real Analysis and Probability . 2002 . Cambridge University Press. 978-0521007542.
- Book: Edgar . Gerald A. . Integral, Probability, and Fractal Measures . 1998 . Springer . 978-1-4419-3112-2.
- Book: Gerald Folland
. Folland . Gerald B. . Gerald Folland. Real Analysis: Modern Techniques and Their Applications . 1999 . Wiley . 0-471-31716-0 . Second.
- Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.
- Book: Fremlin . D.H. . Measure Theory, Volume 2: Broad Foundations . 2016 . Torres Fremlin . Hardback. Second printing.
- Book: Hewitt . Edward . Stromberg . Karl . Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable . 1965 . Springer . 0-387-90138-8.
- R. Duncan Luce and Louis Narens (1987). "measurement, theory of", The , v. 3, pp. 428–32.
- Luther . Norman Y . A decomposition of measures . Canadian Journal of Mathematics . 1967 . 20 . 953–959 . 10.4153/CJM-1968-092-0 . 124262782 . free .
- Book: Mukherjea . A . Pothoven . K . Real and Functional Analysis, Part A: Real Analysis . 1985 . Plenum Press . Second .
- The first edition was published with Part B: Functional Analysis as a single volume: Book: Mukherjea . A . Pothoven . K . Real and Functional Analysis . 1978 . Plenum Press . 10.1007/978-1-4684-2331-0 . 978-1-4684-2333-4 . First .
- M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
- Book: Nielsen . Ole A . An Introduction to Integration and Measure Theory . 1997 . Wiley . 0-471-59518-7.
- Book: Royden . H.L. . Fitzpatrick . P.M. . Halsey Royden . Real Analysis . 2010 . Prentice Hall . 342, Exercise 17.8 . Fourth. First printing. There is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther decomposition) agrees with usual presentations, whereas the first printing's presentation provides a fresh perspective.)
- Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. . Emphasizes the Daniell integral.
- Book: Tao. Terence. Terence Tao. An Introduction to Measure Theory. 2011. American Mathematical Society. Providence, R.I.. 9780821869192.
- Book: Weaver. Nik. Measure Theory and Functional Analysis. 2013. World Scientific. 9789814508568.
External links
Notes and References
- Archimedes Measuring the Circle
- Book: Heath, T. L. . The Works Of Archimedes . 1897 . Cambridge University Press. . Osmania University, Digital Library Of India . 91-98 . Measurement of a Circle.
- 2111.09266 . Bengio . Yoshua . Lahlou . Salem . Deleu . Tristan . Hu . Edward J. . Tiwari . Mo . Bengio . Emmanuel . GFlowNet Foundations . 2021 . cs.LG .
- .
- Book: Bhaskara Rao, K. P. S.. Theory of charges: a study of finitely additive measures. 1983. Academic Press. M. Bhaskara Rao. 0-12-095780-9. London. 35. 21196971.