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This Measure was introduced by Alfréd Haar , a Hungarian Mathematician , about 1932 . Haar measures are used in many parts of analysis and number theory.


PRELIMINARIES


Let ''G'' be a locally compact topological group. In this article, the σ-algebra generated by all Compact Subsets of ''G'' is called the Borel Algebra . An element of the Borel algebra is called a Borel Set .
If ''a'' is an element of ''G'' and ''S'' is a subset of ''G'', then we define the left and right translates of ''S'' as follows:
  • Left translate:

    • : a S = \{a \cdot s: s \in S\}.

  • Right translate:

    • : S a = \{s \cdot a: s \in S\}.


Left and right translates map Borel sets into Borel sets.

A measure μ on the Borel subsets of ''G'' is called ''left-translation-invariant'' if and only if
for all Borel subsets ''S'' of ''G'' and all ''a'' in ''G'' one has
: \mu(a S) = \mu(S). \quad
A similar definition is made for right translation invariance.


EXISTENCE OF THE LEFT HAAR MEASURE


It turns out that there is, Up To a positive multiplicative constant, only one left-translation-invariant countably additive regular measure μ on the Borel subsets of ''G'' such that μ(''U'') > 0 for any open non-empty Borel set ''U''. Here, following Halmos, Section 52, we say μ is regular Iff :

  • μ(''K'') is finite for every compact set ''K''.


  • Every Borel set ''E'' is outer regular:

    • :: \mu(E) = \inf \{\mu(U): E \subseteq U, U \mbox{ open and Borel}\}.


  • If ''E'' is Borel, then ''E'' is inner regular:

    • :: \mu(E) = \sup \{\mu(K): K \subseteq E, K \mbox{ compact }\}.


Remark. In some pathological cases, a set can be open without being Borel. For this reason, in the property of outer regularity, the range of the infimum is specifically stated to be over sets which are open and Borel. These pathologies never occur if ''G'' is a locally compact group whose underlying topology is separable metric; in this case the Borel structure is that generated by all open sets.


THE RIGHT HAAR MEASURE


It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure ν, but it need not coincide with the left-translation-invariant measure μ. These measures are the same only for so-called ''unimodular groups'' (see below). It is quite simple though to find a relationship between μ and ν.

Indeed, for a Borel set ''S'', let us denote by S^{-1} the set of inverses of elements of ''S''. If we define
: \mu_{-1}(S) = \mu(S^{-1}) \quad
then this is a right Haar measure. To show right invariance, apply the definition:

: \mu_{-1}(S a) = \mu((S a)^{-1}) = \mu(a^{-1} S^{-1}) = \mu(S^{-1}) = \mu_{-1}(S). \quad

Because the right measure is unique, it follows that μ-1 is a multiple of ν and so
:\mu(S^{-1})=k
u(S)\,
for all Borel sets ''S'', where ''k'' is some positive constant.


THE HAAR INTEGRAL


Using the general theory of Lebesgue Integration , one can then define an integral for all Borel measurable functions ''f'' on ''G''. This integral is called the Haar integral. If μ is a left Haar measure, then
: \int_G f(s x) \ d\mu(x) = \int_G f(x) \ d\mu(x)
for any integrable function ''f''. This is immediate for step functions, being essentially the definition of left invariance.


USES


The Haar measures are used in Harmonic Analysis on arbitrary locally compact groups, see Pontryagin Duality . A frequently used technique for proving the existence of a Haar measure on a locally compact group ''G'' is showing the existence of a left invariant Radon Measure on ''G''.

Unless ''G'' is a discrete group, it is impossible to define a countably-additive right invariant measure on ''all'' subsets of ''G'', assuming the Axiom Of Choice . See Non-measurable Set s.


EXAMPLES


  • The Haar measure on the topological group (R, +) which takes the value 1 on the interval {Link without Title} is equal to the restriction of Lebesgue Measure to the Borel subsets of ''R''. This can be generalized for (R''n'', +).


  • If ''G'' is the group of positive real numbers with multiplication as operation, then the Haar measure μ(''S'') is given by

    • :: \mu(S) = \int_S rac{1}{t} \, dt

:for any Borel subset ''S'' of the positive reals.

This generalizes to the following:
  • For ''G'' = ''GL''(n,R), left and right Haar measures are proportional and

  More Generally, On Any "http://wwwinformationdelightinfo/encyclopedia/entry/Lie_group" class="copylinks">Lie Group of dimension ''d'' a left Haar measure can be associated with any non-zero left-invariant ''d''-form &omega, as the ''Lebesgue measure'' &omega and similarly for right Haar measures This means also that the modular function can be computed, as the absolute value of the Determinant of the Adjoint Representation