Editing Haar measure (section)

==Examples==

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<li>If <math>G</math> is a [[discrete group]], then the compact subsets coincide with the finite subsets, and a (left and right invariant) Haar measure on <math>G</math> is the [[counting measure]].
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<li>The Haar measure on the topological group <math>(\mathbb{R}, +)</math> that takes the value <math>1</math> on the interval <math>[0,1]</math> is equal to the restriction of [[Lebesgue measure]] to the Borel subsets of <math>\mathbb{R}</math>. This can be generalized to <math>(\mathbb{R}^n, +).</math>
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<li>In order to define a Haar measure <math>\mu</math> on the [[circle group]] <math>\mathbb{T}</math>, consider the function <math>f</math> from <math>[0,2\pi]</math> onto <math>\mathbb{T}</math> defined by <math>f(t)=(\cos(t),\sin(t))</math>. Then <math>\mu</math> can be defined by
<math display="block">\mu(S)=\frac1{2\pi}m(f^{-1}(S)),</math>
where <math>m</math> is the Lebesgue measure on <math>[0,2\pi]</math>. The factor <math>(2\pi)^{-1}</math> is chosen so that <math>\mu(\mathbb{T})=1</math>.
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<li>If <math>G</math> is the group of [[positive real numbers]] under multiplication then a Haar measure <math>\mu</math> is given by
<math display="block"> \mu(S) = \int_S \frac{1}{t} \, dt</math>
for any Borel subset <math>S</math> of positive real numbers.
For example, if <math>S</math> is taken to be an interval <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 <math>[g\cdot a,g\cdot b].</math> Measuring this new interval, we find <math> \mu(gS) = \log((g\cdot b)/(g\cdot a)) = \log(b/a) = \mu(S). </math>
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<li>If <math>G</math> is the group of nonzero real numbers with multiplication as operation, then a Haar measure <math>\mu</math> is given by
<math display="block"> \mu(S) = \int_S \frac{1}{|t|} \, dt </math>
for any Borel subset <math>S</math> of the nonzero reals. 
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<li>For the [[general linear group]] <math>G = GL(n,\mathbb{R})</math>, any left Haar measure is a right Haar measure and one such measure <math>\mu</math> is given by
<math display="block"> \mu(S) = \int_S {1\over |\det(X)|^n} \, dX </math>
where <math>dX</math> denotes the Lebesgue measure on <math>\mathbb{R}^{n^2}</math> identified with the set of all <math>n\times n</math>-matrices.  This follows from the [[change of variables formula]].
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<li>Generalizing the previous three examples, if the group <math>G</math> is represented as an open submanifold of <math>\R^n</math> with [[smooth map|smooth]] group operations, then a left Haar measure on <math>G</math> is given by <math>\frac{1}{|J_{(x\cdot)}(e_1)|}d^n x</math>, where <math>e_1</math> is the group identity element of <math>G</math>, <math>J_{(x\cdot)}(e_1)</math> is the [[Jacobian matrix and determinant|Jacobian determinant]] of left multiplication by <math>x</math> at <math>e_1</math>, and <math>d^n x</math> is the Lebesgue measure on <math>\R^n</math>.  This follows from the [[change of variables formula]]. A right Haar measure is given in the same way, except with <math>J_{(\cdot x)}(e_1)</math> being the Jacobian of right multiplication by <math>x</math>.
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<li>For the [[orthogonal group]] <math>G = O(n)</math>, its Haar measure can be constructed as follows (as the distribution of a random variable). First sample <math>A \sim N(0, 1)^{n\times n}</math>, that is, a matrix with all entries being IID samples of the normal distribution with mean zero and variance one. Next use [[Gram–Schmidt process]] on the matrix; the resulting random variable takes values in <math>O(n)</math> and it is distributed according to the probability Haar measure on that group.<ref>{{Cite journal |last=Diaconis |first=Persi |date=2003-02-12 |title=Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture |journal=Bulletin of the American Mathematical Society |volume=40 |issue=2 |pages=155–178 |doi=10.1090/s0273-0979-03-00975-3 |issn=0273-0979|doi-access=free }}</ref> Since the [[special orthogonal group]] <math>SO(n)</math> is an open subgroup of <math>O(n)</math> the restriction of Haar measure of <math>O(n)</math> to <math>SO(n)</math> gives a Haar measure on <math>SO(n)</math> (in random variable terms this means conditioning the determinant to be 1, an event of probability 1/2). 
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<li>The same method as for <math>O(n)</math> can be used to construct the Haar measure on the [[unitary group]] <math>U(n)</math>. For the [[special unitary group]] <math>G = SU(n)</math> (which has measure 0 in <math>U(n)</math>), its Haar measure can be constructed as follows. First sample <math>A</math> from the Haar measure (normalized to one, so that it's a probability distribution) on <math>U(n)</math>, and let  <math>e^{i\theta} = \det A</math>, where <math>\theta</math> may be any one of the angles, then independently sample <math>k</math> from the uniform distribution on <math>\{1, ..., n\}</math>. Then <math>e^{-i\frac{\theta + 2\pi k}n}A</math> is distributed as the Haar measure on <math>SU(n)</math>. 
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<li>Let <math>G</math> be the set of all affine linear transformations <math>A : \mathbb{R} \to \mathbb{R}</math> of the form <math>r \mapsto x r + y</math> for some fixed <math>x, y \in \mathbb{R}</math> with <math>x > 0.</math> Associate with <math>G</math> the operation of [[function composition]] <math>\circ</math>, which turns <math>G</math> into a non-abelian group. <math>G</math> can be identified with the right half plane <math>(0, \infty) \times \mathbb{R} = \left\{ (x, y) ~:~ x, y \in \mathbb{R}, x > 0 \right\}</math> under which the group operation becomes <math>(s, t) \circ (u, v) = (su, sv + t).</math> A left-invariant Haar measure <math>\mu_L</math> (respectively, a right-invariant Haar measure <math>\mu_R</math>) on <math>G = (0, \infty) \times \mathbb{R}</math> is given by
<math display="block">\mu_L(S) = \int_S \frac{1}{x^2} \,dx\,dy</math> {{space|4}}and{{space|4}} <math display="block">\mu_R(S) = \int_S \frac{1}{x} \,dx\,dy</math>
for any Borel subset <math>S</math> of <math>G = (0, \infty) \times \mathbb{R}.</math> This is because if <math>S \subseteq (0, \infty) \times \mathbb{R}</math> is an open subset then for <math>(s, t) \in G</math> fixed, [[integration by substitution]] gives 
<math display="block">\mu_L((s, t) \circ S) = \int_{(s, t) \circ S} \frac{1}{x^2} \,dx\,dy = \int_{S} \frac{1}{(s u)^2} |(s)(s) - (0)(0)| \,du\,dv = \mu_L(S)</math> 
while for <math>(u, v) \in G</math> fixed, 
<math display="block">\mu_R(S \circ (u, v)) = \int_{S \circ (u, v)} \frac{1}{x} \,dx\,dy = \int_S \frac{1}{s u} |(u)(1) - (v)(0)| \,ds\,dt = \mu_R(S).</math>
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<li>On any [[Lie group]] of dimension <math>d</math> a left Haar measure can be associated with any non-zero left-invariant [[differential form|<math>d</math>-form]] <math>\omega</math>, as the ''Lebesgue measure'' <math>|\omega|</math>; 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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<li>[[File:Hyperbola_E.svg|thumb|right|200px|Shaded area is one square unit.]]
A representation of the Haar measure of positive real numbers in terms of [[area]] under the positive branch of the standard hyperbola ''xy'' = 1 uses Borel sets generated by intervals [''a,b''], ''b'' > ''a'' > 0. For example, ''a'' = 1 and ''b'' = [[Euler’s number]] e  yields and area equal to log (e/1) = 1. Then for any positive real number ''c'' the area over the interval [''ca, cb''] equals log (''b''/''a'') so the area in invariant under multiplication by positive real numbers. Note that the area approaches infinity both as ''a'' approaches zero and ''b'' gets large. Use of this Haar measure to define a logarithm function anchors ''a'' at 1 and considers area over an interval in [b,1], with 0 < ''b'' < 1, as [[negative area]]. In this way the logarithm can take any real value even though measure is always positive or zero.
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<li>If <math>G</math> is the group of non-zero [[quaternion]]s, then <math>G</math> can be seen as an open subset of <math>\R^4</math>. A Haar measure <math>\mu</math> is given by
<math display="block">\mu(S)=\int_S\frac1{(x^2+y^2+z^2+w^2)^2}\,dx\,dy\,dz\,dw</math>
where <math>dx\wedge dy\wedge dz\wedge dw</math> denotes the Lebesgue measure in <math>\mathbb{R}^4</math> and <math>S</math> is a Borel subset of <math>G</math>.
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<li>If <math>G</math> is the additive group of [[p-adic number|<math>p</math>-adic numbers]] for a prime <math>p</math>, then a Haar measure is given by letting <math>a+p^n O</math> have measure <math>p^{-n}</math>, where <math>O</math> is the ring of <math>p</math>-adic integers.
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