In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
, a Borel measure on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is a
measure that is defined on all open sets (and thus on all
Borel sets). Some authors require additional restrictions on the measure, as described below.
Formal definition
Let
be a
locally compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
, and let
be the
smallest σ-algebra that contains the
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s of
; this is known as the σ-algebra of
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s. A Borel measure is any measure
defined on the σ-algebra of Borel sets. A few authors require in addition that
is
locally finite, meaning that
for every
compact set . If a Borel measure
is both
inner regular and
outer regular, it is called a
regular Borel measure. If
is both inner regular, outer regular, and
locally finite, it is called a
Radon measure.
On the real line
The
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
with its
usual topology
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vecto ...
is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case,
is the smallest σ-algebra that contains the open intervals of
. While there are many Borel measures ''μ'', the choice of Borel measure that assigns
for every half-open interval
is sometimes called "the" Borel measure on
. This measure turns out to be the restriction to the Borel σ-algebra of the
Lebesgue measure
In measure theory, a branch of mathematics, 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 wi ...
, which is a
complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the ''completion'' of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and has a
complete measure on it. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e.,
for every Borel measurable set, where
is the Borel measure described above).
Product spaces
If ''X'' and ''Y'' are
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
,
Hausdorff topological spaces, then the set of Borel subsets
of their product coincides with the product of the sets
of Borel subsets of ''X'' and ''Y''. That is, the Borel
functor
:
from the category of second-countable Hausdorff spaces to the category of
measurable spaces preserves finite
products.
Applications
Lebesgue–Stieltjes integral
The
Lebesgue–Stieltjes integral is the ordinary
Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of
bounded variation on the real line. The Lebesgue–Stieltjes measure is a
regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.
Laplace transform
One can define the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
of a finite Borel measure μ on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
by the
Lebesgue integral
:
An important special case is where μ is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
or, even more specifically, the Dirac delta function. In
operational calculus, the Laplace transform of a measure is often treated as though the measure came from a
distribution function ''f''. In that case, to avoid potential confusion, one often writes
:
where the lower limit of 0
− is shorthand notation for
:
This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the
Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the
Laplace–Stieltjes transform.
Hausdorff dimension and Frostman's lemma
Given a Borel measure μ on a metric space ''X'' such that μ(''X'') > 0 and μ(''B''(''x'', ''r'')) ≤ ''r
s'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
dim
Haus(''X'') ≥ ''s''. A partial converse is provided by the
Frostman lemma
In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets.
Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. ...
:
Lemma: Let ''A'' be a
Borel subset of R
''n'', and let ''s'' > 0. Then the following are equivalent:
*''H''
''s''(''A'') > 0, where ''H''
''s'' denotes the ''s''-dimensional
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ...
.
*There is an (unsigned) Borel measure ''μ'' satisfying ''μ''(''A'') > 0, and such that
::
:holds for all ''x'' ∈ R
''n'' and ''r'' > 0.
Cramér–Wold theorem
The
Cramér–Wold theorem in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
states that a Borel
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
on
is uniquely determined by the totality of its one-dimensional projections.
[K. Stromberg, 1994. ''Probability Theory for Analysts''. Chapman and Hall.] It is used as a method for proving joint convergence results. The theorem is named after
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of stat ...
and
Herman Ole Andreas Wold
Herman Ole Andreas Wold (25 December 1908 – 16 February 1992) was a Norwegian-born econometrician and statistician who had a long career in Sweden. Wold was known for his work in mathematical economics, in time series analysis, and in econometri ...
.
References
Further reading
*
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ...
, a finite-dimensional Borel measure
* .
*
*
*
*
Wiener's lemma related
External links
Borel measurea
Encyclopedia of Mathematics
{{DEFAULTSORT:Borel Measure
Measures (measure theory)