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| In [[mathematical analysis]], the '''Haar measure''' is a way to assign an "invariant volume" to subsets of [[locally compact topological group]]s and subsequently define an [[integral]] for functions on those groups.
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| This [[Measure (mathematics)|measure]] was introduced by [[Alfréd Haar]] in 1933.<ref name="Haar">{{Citation | first = A. | last = Haar | author-link = Alfréd Haar | title = Der Massbegriff in der Theorie der kontinuierlichen Gruppen | periodical = [[Annals of Mathematics]] | volume = 34 | series = 2 | issue = 1 | year = 1933 | pages = 147–169 |jstor=1968346}}</ref> Haar measures are used in many parts of [[mathematical analysis|analysis]] and [[number theory]], [[group theory]], [[representation theory]], [[estimation theory]] and [[ergodic theory]].
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| ==Preliminaries==
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| Let (''G,.'') be a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]]. The [[Sigma-algebra|σ-algebra]] generated by all open sets of ''G'' is called the [[Borel algebra]]. An element of the Borel algebra is called a [[Borel set]]. If ''g'' is an element of ''G'' and ''S'' is a subset of ''G'', then we define the left and right [[Coset|translates]] of ''S'' as follows:
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| * Left translate:
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| ::<math> g S = \{g.s\,:\,s \in S\}.</math>
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| * Right translate:
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| ::<math> S g = \{s.g\,:\,s \in S\}.</math>
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| Left and right translates map Borel sets into Borel sets.
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| A measure μ on the Borel subsets of ''G'' is called ''left-translation-invariant'' if for all Borel subsets ''S'' of ''G'' and all ''g'' in ''G'' one has
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| :<math> \mu(g S) = \mu(S). \quad </math>
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| A similar definition is made for right translation invariance.
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| ==Haar's theorem==
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| There is, [[up to]] a positive multiplicative constant, a unique [[countably additive]], nontrivial measure μ on the Borel subsets of ''G'' satisfying the following properties:
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| * The measure μ is left-translation-invariant: μ(''gE'') = μ(''E'') for every ''g'' in ''G'' and Borel set ''E''.
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| * The measure μ is finite on every compact set: μ(''K'') < ∞ for all compact ''K''
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| * The measure μ is [[outer regular]] on Borel sets ''E'':
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| ::<math> \mu(E) = \inf \{\mu(U): E \subseteq U, U \text{ open}\}.</math>
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| * The measure μ is [[inner regular]] on open sets ''E'':
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| ::<math> \mu(E) = \sup \{\mu(K): K \subseteq E, K \text{ compact}\}.</math>
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| Such a measure on ''G'' is called a ''left Haar measure.'' It can be shown as a consequence of the above properties that μ(''U'') > 0 for every non-empty open subset ''U''. In particular, if ''G'' is compact then μ(''G'') is finite and positive, so we can uniquely specify a left Haar measure on ''G'' by adding the normalization condition μ(''G'') = 1.
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| Some authors define a Haar measure on [[Baire set]]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 σ-ring generated by compact sets, and defines Haar measure on these sets.
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| The left Haar measure satisfies the inner regularity condition for all σ-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 with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset {1} x [0,1]. (Compact subsets of this vertical segment are finite sets and points have measure 0, so the measure of any compact subset of this vertical segment is 0. But, using outer regularity, one can show the segment has infinite measure.)
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| The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by [[André Weil]].<ref>{{Citation | last = Weil | first = André | author-link = André Weil | title = L'intégration dans les groupes topologiques et ses applications | series = Actualités Scientifiques et Industrielles | publisher = Hermann | year = 1940 | place = Paris | volume = 869}}''</ref> Weil's proof used the [[axiom of choice]] and [[Henri Cartan]] furnished a proof which avoided its use.<ref>{{Citation | last = Cartan | first = Henri | author-link = Henri Cartan | title = Sur la mesure de Haar | journal = [[Comptes rendus de l'Académie des sciences|Comptes Rendus de l'Académie des Sciences de Paris]] | volume = 211 | pages = 759–762 | year = 1940}}</ref> Cartan's proof also proves the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by Alfsen in 1963.<ref>{{Citation | last = Alfsen | first = E.M. | title = A simplified constructive proof of existence and uniqueness of Haar measure | journal = Math. Scand. | volume = 12 | pages = 106–116 | year = 1963 | url=http://www.mscand.dk/article.php?id=1667}}</ref> The special case of invariant measure for [[Second-countable space|second countable]] locally compact groups had been shown by Haar in 1933.<ref name="Haar"/>
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| ==Construction of Haar measure==
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| ===A construction using compact subsets===
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| The following method of constructing Haar measure is more or less the method used by Haar and Weil.
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| For any subsets ''T'', ''U'' of ''G'' with ''U'' nonempty define [''T'':''U''] to be the smallest number of left translates of ''U'' that cover ''T'' (so this is a non-negative integer or infinity). This is not additive on compact sets ''T'', though it does have the property that [''S'':''U'']+[''T'':''U'']=[''S''∪''T'':''U''] for disjoint compact sets ''S'' and ''T'' provided that ''U'' is a sufficiently small open neighborhood of the identity (depending on ''S'' and ''T''). The idea of Haar measure is to take a sort of limit of [''T'':''U''] as ''U'' becomes smaller to make it additive on all pairs of disjoint compact sets, though it first has to be normalized so that the limit is not just infinity. So fix a compact set ''A'' with non-empty interior (which exists as the group is locally compact) and for a compact set ''T'' define
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| :<math>\mu_A(T)=\lim_U\frac{[T:U]}{[A:U]}</math>
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| where the limit is taken over a suitable directed set of open neighborhoods of the identity eventually contained in any given neighborhood; the existence of a directed set such that the limit exists follows using [[Tychonoff's theorem]].
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| The function μ<sub>''A''</sub> is additive on disjoint compact sets of ''G'', which implies that is is a regular [[content (measure theory)|content]]. From a regular content one can construct a measure by first extending μ<sub>''A''</sub> to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets. (Even for open sets ''T'', the corresponding measure μ<sub>''A''</sub>(''T'') need not be given by the lim sup formula above. The problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact closure, so is not an outer measure.)
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| ===A construction using compactly supported functions===
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| Cartan introduced another way of constructing Haar measure as a [[Radon measure]] (a positive linear functional on compactly supported continuous functions) which is similar to the construction above except that ''A'', ''S'', ''T'', and ''U'' are positive continuous functions of compact support rather than subsets of ''G''. In this case we define [''T'':''U''] to be the inf of numbers ''c''<sub>1</sub>+...+''c''<sub>''n''</sub> such that ''T''(''g'') is less than the linear combination''c''<sub>1</sub>''U''(''g''<sub>1</sub>''g'')+...+''c''<sub>''n''</sub>''U''(''g''<sub>''n''</sub>g) of left translates of ''U'' for some ''g''<sub>1</sub>,...,''g''<sub>''n''</sub>.
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| As before we define
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| :<math>\mu_A(T)=\lim_U\frac{[T:U]}{[A:U]}</math>.
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| The fact that the limit exists takes some effort to prove, though the advantage of doing this is that the proof avoids the use of the axiom of choice and also gives uniqueness of Haar measure as a by-product. The functional μ<sub>A</sub> extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure. (Note that even though the limit is linear in ''T'', the individual terms [''T'':''U''] are not usually linear in ''T''.)
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| ===A construction using mean values of functions===
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| Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups. The idea is that given a function ''f'' on a compact group, one can find a linear combination Σ''a''<sub>''i''</sub>''f''(''g''<sub>''i''</sub>''g'') (where Σ''a''<sub>''i''</sub>=1) of its left translates that differs from a constant function by at most some small number ε. Then one shows that as ε tends to zero the values of these constant functions tend to a limit, which is called the mean value (or integral) of the function ''f''.
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| For groups that are locally compact but not compact this construction does not give Haar measure as the mean value of compactly supported functions is zero. However something like it does work for [[almost periodic function]]s on the group which do have a mean value, though this is not given by with respect to Haar measure.
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| ===A construction on Lie groups===
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| On an ''n''-dimensional Lie group, Haar measure can be constructed easily as the measure induced by a left-invariant ''n''-form. This was known before Haar's theorem.
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| ==The right Haar measure==
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| It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure ν satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure μ. The left and right Haar 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.
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| Indeed, for a Borel set ''S'', let us denote by <math>S^{-1}</math> the set of inverses of elements of ''S''. If we define
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| :<math> \mu_{-1}(S) = \mu(S^{-1}) \quad </math>
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| then this is a right Haar measure. To show right invariance, apply the definition:
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| :<math> \mu_{-1}(S g) = \mu((S g)^{-1}) = \mu(g^{-1} S^{-1}) = \mu(S^{-1}) = \mu_{-1}(S). \quad </math>
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| Because the right measure is unique, it follows that μ<sub>-1</sub> is a multiple of ν and so
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| :<math>\mu(S^{-1})=k\nu(S)\,</math>
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| for all Borel sets ''S'', where ''k'' is some positive constant.
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| ===The modular function===
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| The ''left'' translate of a right Haar measure is a right Haar measure. More precisely, if ν is a right Haar measure, then
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| :<math> S \mapsto \nu (g^{-1} S) \quad </math>
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| is also right invariant. Thus, by uniqueness of the Haar measure, there exists a function Δ from the group to the positive reals, called the '''Haar modulus''', '''modular function''' or '''modular character''', such that for every Borel set ''S''
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| :<math> \nu (g^{-1} S) = \Delta(g) \nu(S). \quad</math>
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| Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation.
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| The modular function is a continuous group homomorphism into the multiplicative group of nonzero real numbers. A group is called '''unimodular''' if the modular function is identically 1, or, equivalently, if the Haar measure is both left and right invariant. Examples of unimodular groups are [[abelian group]]s, [[compact group]]s, [[discrete group]]s (e.g., [[finite group]]s), [[semisimple Lie group]]s and connected [[nilpotent Lie group]]s. An example of a non-unimodular group is the group of affine transformations
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| :<math>\big\{ x \mapsto a x + b : a\in\R\setminus\{0\}, b\in\R \big\}=\big\{\begin{bmatrix}
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| a & b \\ \\
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| 0 & 1 \end{bmatrix}\big\}</math>
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| on the real line. This example shows that a solvable Lie group need not be unimodular.
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| In this group a left Haar measure is given by ''dadb''/''a''<sup>2</sup>, and a right Haar measure by ''dadb''/|''a''|.
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| ==Measures on homogeneous spaces==
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| If the locally compact group ''G'' acts transitively on a space ''G''/''H'', one can ask if this space has an invariant measure, or more generally a relatively invariant measure with the property that μ(''gE'') = χ(''g'')μ(''E'') for some character χ of ''G''. A necessary and sufficient condition for the existence of such a measure is that χ=Δ/δ on ''H'', where Δ and δ are the modular functions of ''G'' and ''H''.
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| In particular an invariant measure on ''Q'' exists if and only if the modular function of ''G'' restricted to ''H'' is the modular function of ''H''.
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| Example. If ''G'' is the group ''SL''<sub>2</sub>('''R''') and ''H'' the subgroup of upper triangular matrices, then the modular function of ''H'' is nontrivial but the modular function of ''G'' is trivial. The quotient of these cannot be extended to any character of ''G'', so the quotient space ''G''/''H'' (which can be thought of as 1-dimensional real projective space) does not even have a relatively invariant measure.
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| ==Haar integral==
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| {{expand section|date=December 2011}}
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| 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
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| :<math> \int_G f(sx) \ d\mu(x) = \int_G f(x) \ d\mu(x) </math>
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| for any integrable function ''f''. This is immediate for [[indicator function]]s, being essentially the definition of left invariance.
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| ==Examples==
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| * A Haar measure on the topological group ('''R''', +) which takes the value 1 on the interval [0,1] is equal to the restriction of [[Lebesgue measure]] to the Borel subsets of '''R'''. This can be generalized to ('''R'''<sup>''n''</sup>, +).
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| * If ''G'' is the group of nonzero real numbers with multiplication as operation, then a Haar measure μ is given by
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| ::<math> \mu(S) = \int_S \frac{1}{|t|} \, dt </math>
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| :for any Borel subset ''S'' of the nonzero reals.
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| :For example, if <math>S</math> is taken to be an interval between two points <math>a,b</math>, then we find <math> \mu(S) = \log(b/a) </math>. Now we let the multiplicative group act on this interval by a multiplication of all its elements by a number <math>g</math>, resulting in <math>gS</math> being the interval with bounds <math>g\cdot a,g\cdot b</math>. Measuring this new integral, we find <math> \mu(gS) = \log((g\cdot b)/(g\cdot a)) = \log(b/a) = \mu(S) </math>.
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| * If the group ''G'' is represented as an open submanifold of '''R'''<sup>''n''</sup> then a left Haar measure on ''G'' is given by ''d''<sup>''n''</sup>''x''/''J''(''x''), where ''J''(''x'') is the Jacobian of left multiplication by ''x''. A right Haar measure is given in the same way, except with ''J''(''x'') the Jacobian of right multiplication by ''x''.
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| * As a special case of the previous construction, for ''G'' = ''GL''(n,'''R'''), any left Haar measure is a right Haar measure and one such measure μ is given by
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| ::<math> \mu(S) = \int_S {1\over |\det(X)|^n} \, dX </math>
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| :where ''dX'' denotes the Lebesgue measure on <math>\mathbb{R}^{n^2}</math>, the set of all <math>n\times n</math>-matrices. This follows from the [[change of variables formula]].
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| * On any [[Lie group]] of dimension ''d'' a left Haar measure can be associated with any non-zero left-invariant ''d''-form ω, as the ''Lebesgue measure'' |ω|; 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 of a Lie group|adjoint representation]].
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| * In order to define a Haar measure μ on the unit circle ''T'', consider the function ''f'' from [0,2π] onto ''T'' defined by ''f''(''t'') = (cos(''t''),sin(''t'')). Then μ can be defined by
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| ::<math>\mu(S)=\frac1{2\pi}m\left(f^{-1}(S)\right),</math>
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| :where ''m'' is the Lebesgue measure. The factor (2π)<sup>−1</sup> is chosen so that μ(''T'') = 1.
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| * If ''G'' is the group of non-null [[quaternion]]s, then ''G'' can be seen as an open subset of '''R'''<sup>4</sup>. A Haar measure μ is given by
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| ::<math>\mu(S)=\int_S\frac1{(x^2+y^2+z^2+w^2)^2}\,dx\,dy\,dz\,dw</math>
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| :where ''dx dy dz dw'' denotes the Lebesgue measure in '''R'''<sup>4</sup> and ''S'' is a Borel subset of ''G''.
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| * If ''G'' is the additive group of ''p''-adic numbers for a prime ''p'', then a Haar measure is given by letting ''a''+''p''<sup>''n''</sup>''O'' have measure ''p''<sup>−''n''</sup>, where ''O'' is the ring of ''p''-adic integers.
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| ==Uses==
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| Historically, the first use of the Haar theorem was the solution, by [[John von Neumann|von Neumann]], of [[Hilbert's fifth problem]] in the case of compact groups.<ref>{{Citation | first = J. | last = von Neumann | author-link = John von Neumann | title = Die Einfuhrung Analytischer Parameter in Topologischen Gruppen | periodical = [[Annals of Mathematics]] | volume = 34 | issue = 1 | series = 2 | year = 1933 | pages = 170–179 |jstor=1968347}}</ref> In fact, von Neumann's article was published in the same issue of [[Annals of Mathematics]] as Haar's article and immediately after it.
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| 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''.
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| In [[estimation theory]], Haar measures can be used as [[non-informative prior]]s, being [[Jeffreys prior]]s for various questions. For instance, translation invariance of the ([[improper prior|improper]]) uniform distribution on the real numbers (the Haar measure with respect to addition) corresponds to no information about location, and thus it is the Jeffreys prior for the unknown mean of a Gaussian distribution, the mean being a measure of location.
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| Unless ''G'' is a discrete group, it is impossible to define a countably-additive left invariant regular measure on ''all'' subsets of ''G'', assuming the [[axiom of choice]]. See [[non-measurable set]]s.
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| ==Weil's converse theorem==
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| In 1936 Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure for which one can define a convolution product, then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as Haar measure on this completion.
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| ==See also==
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| *[[Pontryagin duality]]
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| ==Notes==
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| {{Reflist}}
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| ==Further reading==
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| * Lynn Loomis, ''An Introduction to Abstract Harmonic Analysis'', D. van Nostrand and Co., 1953.
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| *{{citation|mr=0156915|last1=Hewitt|first1= Edwin|last2=Ross|first2= Kenneth A.
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| |title=Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations.
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| |series=Die Grundlehren der mathematischen Wissenschaften|volume= 115 |publisher= Springer-Verlag|place= Berlin-Göttingen-Heidelberg|year= 1963}}
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| * {{Citation | last = Nachbin | first = Leopoldo | authorlink = Leopoldo Nachbin | title = The Haar Integral | publisher = D. Van Nostrand | location = Princeton, NJ | year = 1965 }}
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| * [[André Weil]], ''Basic Number Theory'', Academic Press, 1971.
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| ==External links==
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| * [http://www.math.ku.dk/kurser/2004-2/mat3re/haarintegral.pdf The existence and uniqueness of the Haar integral on a locally compact topological group] - by Gert K. Pedersen
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| * [http://www.artofproblemsolving.com/LaTeX/Examples/HaarMeasure.pdf On the Existence and Uniqueness of Invariant Measures on Locally Compact Groups] - by Simon Rubinstein-Salzedo
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| {{integral}}
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| {{DEFAULTSORT:Haar Measure}}
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| [[Category:Lie groups]]
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| [[Category:Topological groups]]
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| [[Category:Measures (measure theory)]]
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| [[Category:Fourier analysis]]
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