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Lebesgue measure
From Wikipedia, the free encyclopedia
Jump to: navigation, search
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue,
is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n =
1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also
called n-dimensional volume, n-volume, or simply volume.[1] It is used throughout real
analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue
measure are called Lebesgue measurable; the measure of the Lebesgue measurable set A is
denoted by λ(A).
Henri Lebesgue described this measure in the year 1901, followed the next year by his
description of the Lebesgue integral. Both were published as part of his dissertation in 1902.[2]
The Lebesgue measure is often denoted dx, but this should not be confused with the distinct
notion of a volume form.
1 Examples
2 Properties
3 Null sets
4 Construction of the Lebesgue measure
5 Relation to other measures
6 See also
7 References
[edit] Examples
Any closed interval [a, b] of real numbers is Lebesgue measurable, and its Lebesgue
measure is the length b−a. The open interval (a, b) has the same measure, since the
difference between the two sets consists only of the end points a and b and has measure
Any Cartesian product of intervals [a, b] and [c, d] is Lebesgue measurable, and its
Lebesgue measure is (b−a)(d−c), the area of the corresponding rectangle.
The Lebesgue measure of the set of rational numbers in an interval of the line is 0,
although the set is dense in the interval.
The Cantor set is an example of an uncountable set that has Lebesgue measure zero.
Vitali sets are examples of sets that are not measurable with respect to the Lebesgue
measure. Their existence relies on the axiom of choice.
[edit] Properties
The Lebesgue measure on Rn has the following properties:
1. If A is a cartesian product of intervals I1 × I2 × ... × In, then A is Lebesgue measurable and
Here, |I| denotes the length of the interval I.
2. If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is
itself Lebesgue measurable and λ(A) is equal to the sum (or infinite series) of the
measures of the involved measurable sets.
3. If A is Lebesgue measurable, then so is its complement.
4. λ(A) ≥ 0 for every Lebesgue measurable set A.
5. If A and B are Lebesgue measurable and A is a subset of B, then λ(A) ≤ λ(B). (A
consequence of 2, 3 and 4.)
6. Countable unions and intersections of Lebesgue measurable sets are Lebesgue
measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under
complements and disjoint countable unions need not be closed under countable unions:
7. If A is an open or closed subset of Rn (or even Borel set, see metric space), then A is
Lebesgue measurable.
8. If A is a Lebesgue measurable set, then it is "approximately open" and "approximately
closed" in the sense of Lebesgue measure (see the regularity theorem for Lebesgue
9. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
10. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the
whole of Rn.
11. If A is a Lebesgue measurable set with λ(A) = 0 (a null set), then every subset of A is also
a null set. A fortiori, every subset of A is measurable.
12. If A is Lebesgue measurable and x is an element of Rn, then the translation of A by x,
defined by A + x = {a + x : a ∈ A}, is also Lebesgue measurable and has the same
measure as A.
13. If A is Lebesgue measurable and
, then the dilation of by defined by
is also Lebesgue measurable and has measure
14. More generally, if T is a linear transformation and A is a measurable subset of Rn, then
T(A) is also Lebesgue measurable and has the measure
All the above may be succinctly summarized as follows:
The Lebesgue measurable sets form a σ-algebra containing all products of intervals, and
λ is the unique complete translation-invariant measure on that σ-algebra with
The Lebesgue measure also has the property of being σ-finite.
[edit] Null sets
Main article: Null set
A subset of Rn is a null set if, for every ε > 0, it can be covered with countably many products of
n intervals whose total volume is at most ε. All countable sets are null sets.
If a subset of Rn has Hausdorff dimension less than n then it is a null set with respect to ndimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on
Rn (or any metric Lipschitz equivalent to it). On the other hand a set may have topological
dimension less than n and have positive n-dimensional Lebesgue measure. An example of this is
the Smith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional
Lebesgue measure.
In order to show that a given set A is Lebesgue measurable, one usually tries to find a "nicer" set
B which differs from A only by a null set (in the sense that the symmetric difference (A − B) (B
− A) is a null set) and then show that B can be generated using countable unions and intersections
from open or closed sets.
[edit] Construction of the Lebesgue measure
The modern construction of the Lebesgue measure is an application of Carathéodory's extension
theorem. It proceeds as follows.
Fix n ∈ N. A box in Rn is a set of the form
where bi ≥ ai, and the product symbol here represents a Cartesian product. The volume vol(B) of
this box is defined to be
For any subset A of Rn, we can define its outer measure λ*(A) by:
We then define the set A to be Lebesgue measurable if for every subset S of Rn,
These Lebesgue measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(A)
= λ*(A) for any Lebesgue measurable set A.
The existence of sets that are not Lebesgue measurable is a consequence of a certain settheoretical axiom, the axiom of choice, which is independent from many of the conventional
systems of axioms for set theory. The Vitali theorem, which follows from the axiom, states that
there exist subsets of R that are not Lebesgue measurable. Assuming the axiom of choice, nonmeasurable sets with many surprising properties have been demonstrated, such as those of the
Banach–Tarski paradox.
In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue measurable
is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom
of choice (see Solovay's model).[3]
[edit] Relation to other measures
The Borel measure agrees with the Lebesgue measure on those sets for which it is defined;
however, there are many more Lebesgue-measurable sets than there are Borel measurable sets.
The Borel measure is translation-invariant, but not complete.
The Haar measure can be defined on any locally compact group and is a generalization of the
Lebesgue measure (Rn with addition is a locally compact group).
The Hausdorff measure is a generalization of the Lebesgue measure that is useful for measuring
the subsets of Rn of lower dimensions than n, like submanifolds, for example, surfaces or curves
in R³ and fractal sets. The Hausdorff measure is not to be confused with the notion of Hausdorff
It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.
[edit] See also
Lebesgue's density theorem
[edit] References
1. ^ The term volume is also used, more strictly, as a synonym of 3-dimensional volume
2. ^ Henri Lebesgue (1902). Intégrale, longueur, aire. Université de Paris.
3. ^ Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue
measurable". Annals of Mathematics. Second Series 92 (1): 1–56. doi:10.2307/1970696.
JSTOR 1970696.
Complete measure
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article needs additional citations for verification. Please help improve this
article by adding citations to reliable sources. Unsourced material may be challenged
and removed. (October 2010)
In mathematics, a complete measure (or, more precisely, a complete measure space) is a
measure space in which every subset of every null set is measurable (having measure zero).
More formally, (X, Σ, μ) is complete if and only if
1 Motivation
2 Construction of a complete measure
3 Examples
4 References
[edit] Motivation
The need to consider questions of completeness can be illustrated by considering the problem of
product spaces.
Suppose that we have already constructed Lebesgue measure on the real line: denote this
measure space by (R, B, λ). We now wish to construct two-dimensional Lebesgue measure λ2 on
the plane R2 as a product measure. Naïvely, we would take the σ-algebra on R2 to be B ⊗ B, the
smallest σ-algebra containing all measurable "rectangles" A1 × A2 for Ai ∈ B.
While this approach does define a measure space, it has a flaw. Since every singleton set has
one-dimensional Lebesgue measure zero,
for "any" subset A of R. However, suppose that A is a non-measurable subset of the real line,
such as the Vitali set. Then the λ2-measure of {0} × A is not defined, but
and this larger set does have λ2measure zero. So, "two-dimensional Lebesgue measure" as just
defined is not complete, and some kind of completion procedure is required.
[edit] Construction of a complete measure
Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ0, μ0) of this
measure space that is complete. The smallest such extension (i.e. the smallest σ-algebra Σ0) is
called the completion of the measure space.
The completion can be constructed as follows:
let Z be the set of all subsets of μ-measure zero subsets of X (intuitively, those elements
of Z that are not already in Σ are the ones preventing completeness from holding true);
let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains
every element of Σ and of Z);
there is a unique extension μ0 of μ to Σ0 given by the infimum
Then (X, Σ0, μ0) is a complete measure space, and is the completion of (X, Σ, μ).
In the above construction it can be shown that every member of Σ0 is of the form A ∪ B for some
A ∈ Σ and some B ∈ Z, and
[edit] Examples
Borel measure as defined on the Borel σ-algebra generated by the open intervals of the
real line is not complete, and so the above completion procedure must be used to define
the complete Lebesgue measure.
n-dimensional Lebesgue measure is the completion of the n-fold product of the onedimensional Lebesgue space with itself. It is also the completion of the Borel measure, as
in the one-dimensional case.
[edit] References
Terekhin, A.P. (2001), "Complete measure", in Hazewinkel, Michiel, Encyclopedia of
Mathematics, Springer, ISBN 978-1556080104,
Hausdorff dimension
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Estimating the Hausdorff dimension of the coast of Great Britain
In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch
dimension) is an extended non-negative real number associated with any metric space. The
Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the
Hausdorff dimension of an n-dimensional inner product space equals n. This means, for example,
the Hausdorff dimension of a point is zero, the Hausdorff dimension of a line is one, and the
Hausdorff dimension of the plane is two. There are, however, many irregular sets that have
noninteger Hausdorff dimension. The concept was introduced in 1918 by the mathematician
Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension
for highly irregular sets were obtained by Abram Samoilovitch Besicovitch.
Sierpinski triangle. A space with fractal dimension log 3 / log 2, which is approximately 1.5849625
1 Intuition
2 Formal definition
3 Examples
4 Properties of Hausdorff dimension
o 4.1 Hausdorff dimension and inductive dimension
o 4.2 Hausdorff dimension and Minkowski dimension
o 4.3 Hausdorff dimensions and Frostman measures
o 4.4 Behaviour under unions and products
5 Self-similar sets
6 The Hausdorff dimension theorem
7 See also
8 Historical references
9 Notes
10 References
[edit] Intuition
The intuitive dimension of a geometric object is the number of independent parameters you need
to pick out a unique point inside. But you can easily take a single real number, one parameter,
and split its digits to make two real numbers. The example of a space-filling curve shows that
you can even take one real number into two continuously, so that a one-dimensional object can
completely fill up a higher dimensional object.
Every space filling curve hits every point many times, and does not have a continuous inverse. It
is impossible to map two dimensions onto one in a way that is continuous and continuously
invertible. The topological dimension explains why. The Lebesgue covering dimension is
defined as the minimum number of overlaps that small open balls need to have in order to
completely cover the object. When you try to cover a line by dropping open intervals on it, you
always end up covering some points twice. Covering a plane with disks, you end up covering
some points three times, etc. The topological dimension tells you how many different little balls
connect a given point to other points in the space, generically. It tells you how difficult it is to
break a geometric object apart into pieces by removing slices.
But the topological dimension doesn't tell you anything about volumes. A curve which is almost
space filling can still have topological dimension one, even if it fills up most of the area of a
region. A fractal has an integer topological dimension, but in terms of the amount of space it
takes up, it behaves as a higher dimensional space. The Hausdorff dimension defines the size
notion of dimension, which requires a notion of radius, or metric.
Consider the number N(r) of balls of radius at most r required to cover X completely. When r is
small, N(r) is large. If N(r) always grows as 1/rd as r approaches zero, then X has Hausdorff
dimension d. The precise definition requires that the dimension "d" so defined is a critical
boundary between growth rates that are insufficient to cover the space, and growth rates that are
For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional
geometry and science, the Hausdorff dimension is an integer. But Benoît Mandelbrot observed
that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He
observed that the proper idealization of most rough shapes you see around you is not in terms of
smooth idealized shapes, but in terms of fractal idealized shapes:
clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not
smooth, nor does lightning travel in a straight line. [1]
The Hausdorff dimension is a successor to the less sophisticated but in practice very similar boxcounting dimension or Minkowski–Bouligand dimension. This counts the squares of graph paper
in which a point of X can be found as the size of the squares is made smaller and smaller. For
fractals that occur in nature, the two notions coincide. The packing dimension is yet another
similar notion. These notions (packing dimension, Hausdorff dimension, Minkowski–Bouligand
dimension) all give the same value for many shapes, but there are well documented exceptions.
[edit] Formal definition
be a metric space. If
of is defined by
In other words,
, the
-dimensional Hausdorff content
is the infimum of the set of numbers
(indexed) collection of balls
which satisfies
Hausdorff dimension of
such that there is some
for each
. (Here, we use the standard convention that inf Ø =∞.) The
is defined by
may be defined as the infimum of the set of
such that the
-dimensional Hausdorff measure of
is zero. This is the same as the supremum of the set of
such that the -dimensional Hausdorff measure of
is infinite (except that when
this latter set of numbers is empty the Hausdorff dimension is zero).
[edit] Examples
The Euclidean space
has Hausdorff dimension n.
The circle S1 has Hausdorff dimension 1.
Countable sets have Hausdorff dimension 0.
Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension.
For example, the Cantor set (a zero-dimensional topological space) is a union of two copies of
itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff
dimension is
which is approximately
The Sierpinski triangle is a union of three
copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of
, which is approximately
Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension
as the space they fill.
The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost
An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension and subsequent work by other authors have claimed that
the Hausdorff dimension of many coastlines can be estimated. Their results have varied from
1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. However,
'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and
cannot be regarded rigorously as a Hausdorff dimension. It is based on scaling properties of
coastlines at a large range of scales; however, it does not include all arbitrarily small scales,
where measurements would depend on atomic and sub-atomic structures, and are not well
The bond system of an amorphous solid changes its Hausdorff dimension from Euclidian 3 below
glass transition temperature Tg (where the amorphous material is solid), to fractal 2.55±0.05
above Tg, where the amorphous material is liquid.[2]
[edit] Properties of Hausdorff dimension
[edit] Hausdorff dimension and inductive dimension
Let X be an arbitrary separable metric space. There is a topological notion of inductive
dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted
Theorem. Suppose X is non-empty. Then
where Y ranges over metric spaces homomorphic to X. In other words, X and Y have the same
underlying set of points and the metric dY of Y is topologically equivalent to dX.
These results were originally established by Edward Szpilrajn (1907–1976). The treatment in
Chapter VII of the Hurewicz and Wallman reference is particularly recommended.
[edit] Hausdorff dimension and Minkowski dimension
The Minkowski dimension is similar to the Hausdorff dimension, except that it is not associated
with a measure. The Minkowski dimension of a set is at least as large as the Hausdorff
dimension. In many situations, they are equal. However, the set of rational points in
Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which
the Minkowski dimension is strictly larger than the Hausdorff dimension.
[edit] Hausdorff dimensions and Frostman measures
If there is a measure
defined on Borel subsets of a metric space
holds for some constant
such that
and for every ball
, then
. A partial converse is provided by Frostman's lemma. That article also
discusses another useful characterization of the Hausdorff dimension.
[edit] Behaviour under unions and products
is a finite or countable union, then
This can be verified directly from the definition.
are metric spaces, then the Hausdorff dimension of their product satisfies[3]
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has
dimension 1.[4] In the opposite direction, it is known that when
and are Borel subsets of
, the Hausdorff dimension of
is bounded from above by the Hausdorff dimension of
plus the upper packing dimension of . These facts are discussed in Mattila (1995).
[edit] Self-similar sets
Many sets defined by a self-similarity condition have dimensions which can be determined
explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ,
that is ψ(E) = E, although the exact definition is given below.
Theorem. Suppose
are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique nonempty compact set A such that
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to
the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance.[5]
To determine the dimension of the self-similar set A (in certain cases), we need a technical
condition called the open set condition on the sequence of contractions ψi which is stated as
follows: There is a relatively compact open set V such that
where the sets in union on the left are pairwise disjoint.
Theorem. Suppose the open set condition holds and each ψi is a similitude, that is a composition
of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose
Hausdorff dimension is s where s is the unique solution of [6]
Note that the contraction coefficient of a similitude is the magnitude of the dilation.
We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or
sometimes called Sierpinski gasket). Consider three non-collinear points a1, a2, a3 in the plane R²
and let ψi be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the
corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of
Taking natural logarithms of both sides of the above equation, we can solve for s, that is:
The Sierpinski gasket is self-similar. In general a set E which is a fixed point of a mapping
is self-similar if and only if the intersections
where s is the Hausdorff dimension of E and
denotes Hausdorff measure. This is clear in the
case of the Sierpinski gasket (the intersections are just points), but is also true more generally:
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is
[edit] The Hausdorff dimension theorem
The following theorem deals with existence of fractals with given Hausdorff dimension in
Euclidean spaces [7]:
Theorem. For any real
and integer
dimension in -dimensional Euclidean space.
, there is a continuum fractals with Hausdorff
[edit] See also
List of fractals by Hausdorff dimension Examples of deterministic fractals, random and natural
Intrinsic dimension
[edit] Historical references
A. S. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions". Mathematische
Annalen 101 (1): 161–193. doi:10.1007/BF01454831.
A. S. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions". Journal of the London
Mathematical Society 12 (1): 18–25. doi:10.1112/jlms/s1-12.45.18.
Several selections from this volume are reprinted in Edgar, Gerald A. (1993). Classics on fractals.
Boston: Addison-Wesley. ISBN 0-201-58701-7. See chapters 9,10,11
F. Hausdorff (March 1919). "Dimension und äußeres Maß". Mathematische Annalen 79 (1–2):
157–179. doi:10.1007/BF01457179.
[edit] Notes
1. ^ Mandelbrot, Benoît (1982). The Fractal Geometry of Nature. Lecture notes in mathematics
1358. W. H. Freeman. ISBN 0716711869.
2. ^ M.I. Ojovan, W.E. Lee. (2006). "Topologically disordered systems at the glass transition". J.
Phys.: Condensed Matter 18 (50): 11507–20. Bibcode 2006JPCM...1811507O. doi:10.1088/09538984/18/50/007.
3. ^ Marstrand, J. M. (1954). "The dimension of Cartesian product sets". Proc. Cambridge Philos.
Soc. 50 (3): 198–202. doi:10.1017/S0305004100029236.
4. ^ Falconer, Kenneth J. (2003). Fractal geometry. Mathematical foundations and applications.
John Wiley & Sons, Inc., Hoboken, New Jersey.
5. ^ Falconer, K. J. (1985). "Theorem 8.3". The Geometry of Fractal Sets. Cambridge, UK: Cambridge
University Press. ISBN 0-521-25694-1.
6. ^ Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically". Phys. Rev.
Lett. 57 (12): 1390–1393. doi:10.1103/PhysRevLett.57.1390. PMID 10033437.
7. ^ Soltanifar, M.(2006) On A Sequence of Cantor Fractals, Rose Hulman Undergraduate
Mathematics Journal, Vol 7, No 1, paper 9.
[edit] References
Dodson, M. Maurice; Kristensen, Simon (June 12, 2003). "Hausdorff Dimension and Diophantine
Approximation". Fractal geometry and applications: a jubilee of Beno\^it Mandelbrot. Part, -347, Proc. Sympos. Pure Math., 72, Part , Amer. Math. Soc., Providence, RI, . 1 (305).
Hurewicz, Witold; Wallman, Henry (1948). Dimension Theory. Princeton University Press.
E. Szpilrajn (1937). "La dimension et la mesure". Fundamenta Mathematica 28: 81–9.
Marstrand, J. M. (1954). "The dimension of cartesian product sets". Proc. Cambridge Philos. Soc.
50 (3): 198–202. doi:10.1017/S0305004100029236.
Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces. Cambridge University
Press. ISBN 978-0-521-65595-8.
Carathéodory's extension theorem
From Wikipedia, the free encyclopedia
Jump to: navigation, search
See also Carathéodory's theorem for other meanings.
In measure theory, Carathéodory's extension theorem (named after the Greek mathematician
Constantin Carathéodory) states that any σ-finite measure defined on a given ring R of subsets of
a given set Ω can be uniquely extended to the σ-algebra generated by R. Consequently, any
measure on a space containing all intervals of real numbers can be extended to the Borel algebra
of the set of real numbers. This is an extremely powerful result of measure theory, and proves for
example the existence of the Lebesgue measure.
1 Semi-ring and ring
o 1.1 Definitions
o 1.2 Properties
o 1.3 Motivation
o 1.4 Example
2 Statement of the theorem
3 See also
4 References
[edit] Semi-ring and ring
[edit] Definitions
For a given set Ω, we may define a semi-ring as a subset S of
has the following properties:
, the power set of Ω, which
For all A, B ∈ S, we have A∩B ∈ S (closed under pairwise intersections)
For all A, B ∈ S, there exist disjoint sets Ki ∈ S, with i = 1, 2, …, n, such that
(relative complements can be written as finite disjoint unions).
With the same notation, we define a ring R as a subset of the power set of Ω which has the
following properties:
For all A, B ∈ R, we have A∪B ∈ R (closed under pairwise unions)
For all A, B ∈ R, we have A\B ∈ R (closed under relative complements).
Thus any ring on Ω is also a semi-ring.
Sometimes, the following constraint is added in the measure theory context:
Ω is the disjoint union of a countable family of sets in S.
[edit] Properties
Arbitrary (possibly uncountable) intersections of rings on Ω are still rings on Ω.
If A is a non-empty subset of
, then we define the ring generated by A (noted R(A)) as the
smallest ring containing A. It is straightfoward to see that the ring generated by A is equivalent
to the intersection of all rings containing A.
For a semi-ring S, the set containing all finite disjoint union of sets of S is the ring generated by S:
(R(S) is simply the set containing all finite unions of sets in S).
A content μ defined on a semi-ring S can be extended on the ring generated by S. Such an
extension is unique. The extended content can be written:
, with the Ap in S.
In addition, it can be proved that μ is a pre-measure if and only if the extended content is also a
pre-measure, and that any pre-measure on R(S) that extends the pre-measure on S is necessarily
of this form.
[edit] Motivation
In measure theory, we are not interested in semi-rings and rings themselves, but rather in σalgebras generated by them. The idea is that it is possible to build a pre-measure on a semi-ring S
(for example Stieltjes measures), which can then be extended to a pre-measure on R(S), which
can finally be extended to a measure on a σ-algebra through Caratheodory's extension theorem.
As σ-algebras generated by semi rings and rings are the same, the difference does not really
matter (in the measure theory context at least). Actually, the Carathéodory's extension theorem
can be slightly generalized by replacing ring by semi-ring.
The definition of semi-ring may seem a bit convoluted, but the following example shows why it
is useful.
[edit] Example
Think about the subset of
defined by the set of all half-open intervals [a, b) for a and b
reals. This is a semi-ring, but not a ring. Stieltjes measures are defined on intervals; the countable
additivity on the semi ring is not too difficult to prove because we only consider countable
unions of intervals which are intervals themselves. Proving it for arbitrary countably union of
intervals is proved using Caratheodory's theorem.
[edit] Statement of the theorem
Let R be a ring on Ω and μ: R → [0, + ∞] be a pre-measure on a R.
The Carathéodory's extension theorem states that[1] there exists a measure μ′: σ(R) → [0, + ∞]
such that μ′ is an extension of μ. (That is, μ′ |R = μ).
Here σ(R) is the σ-algebra generated by R.
If μ is σ-finite then the extension μ′ is unique (and also σ-finite)[2].
[edit] See also
Outer measure: the proof of Carathéodory's extension theorem is based upon the outer
measure concept.
Hahn-Kolmogorov theorem
Loeb measures, constructed using Carathéodory's extension theorem.
[edit] References
1. ^ Vaillant, Theorem 4
2. ^ Ash, p19
Noel Vaillant, Caratheodory's Extension, on A clear demonstration of the
theorem through exercises.
Robert B. Ash (1999). Probability and Measure theory. Academic Press; 2 edition.
ISBN 0120652021.