In
mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of
locally compact topological groups, consequently defining an
integral for functions on those groups.
This
measure was introduced by
Alfréd Haar in 1933, though its special case for
Lie groups 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 died ...
in 1897 under the name "invariant integral".
Haar measures are used in many parts of
analysis,
number theory,
group theory,
representation theory,
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
,
probability theory, 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 ev ...
Hausdorff topological group. The
-algebra generated by all open subsets of
is called the
Borel algebra. An element of the Borel algebra is called a
Borel set. If
is an element of
and
is a subset of
, then we define the left and right
translates
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
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 object, 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'' wi ...
a positive multiplicative constant, a unique
countably additive, 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 on Borel sets
:
* The measure
is
inner regular 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 sets 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 (with its usual topology) and the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), 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 ...
with the
discrete topology is a locally compact group with the
product topology and a Haar measure on this group is not inner regular for the closed subset