In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, the Haar measure assigns an "invariant volume" to subsets of
locally compact topological group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
s, consequently defining an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
for functions on those groups.
This
measure was introduced by
Alfréd Haar
Alfréd Haar ( hu, Haar Alfréd; 11 October 1885, Budapest – 16 March 1933, Szeged) was a Hungarian mathematician. In 1904 he began to study at the University of Göttingen. His doctorate was supervised by David Hilbert. The Haar mea ...
in 1933, though its special case for
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
had been introduced by
Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and d ...
in 1897 under the name "invariant integral".
Haar measures are used in many parts of
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
,
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
,
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
,
statistics
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
,
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, and
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
.
Preliminaries
Let
be a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
Hausdorff topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
. The
-algebra generated by all open subsets of
is called the
Borel algebra
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 named ...
. An element of the Borel algebra is called a
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 name ...
. If
is an element of
and
is a subset of
, then we define the left and right
translates of
by ''g'' as follows:
* Left translate:
* Right translate:
Left and right translates map Borel sets onto Borel sets.
A measure
on the Borel subsets of
is called ''left-translation-invariant'' if for all Borel subsets
and all
one has
:
A measure
on the Borel subsets of
is called ''right-translation-invariant'' if for all Borel subsets
and all
one has
:
Haar's theorem
There is,
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
a positive multiplicative constant, a unique
countably additive
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivit ...
, nontrivial measure
on the Borel subsets of
satisfying the following properties:
* The measure
is left-translation-invariant:
for every
and all Borel sets
.
* The measure
is finite on every compact set:
for all compact
.
* The measure
is
outer regular
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Bo ...
on Borel sets
:
* The measure
is
inner regular
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.
Definition
Let (''X'', ''T'') be a Hausdorff topological space and let Σ be a σ-algebra on ''X'' th ...
on open sets
:
Such a measure on
is called a ''left Haar measure.'' It can be shown as a consequence of the above properties that
for every non-empty open subset
. In particular, if
is compact then
is finite and positive, so we can uniquely specify a left Haar measure on
by adding the normalization condition
.
In complete analogy, one can also prove the existence and uniqueness of a ''right Haar measure'' on
. The two measures need not coincide.
Some authors define a Haar measure on
Baire set
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets.
There are several inequivalent definitions of Baire sets, but in the mo ...
s rather than Borel sets. This makes the regularity conditions unnecessary as Baire measures are automatically regular.
Halmos rather confusingly uses the term "Borel set" for elements of the
-ring generated by compact sets, and defines Haar measures on these sets.
The left Haar measure satisfies the inner regularity condition for all
-finite Borel sets, but may not be inner regular for ''all'' Borel sets. For example, the product of the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
(with its usual topology) and 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 poin ...
with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
is a locally compact group with the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
and a Haar measure on this group is not inner regular for the closed subset