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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. The modular function is a continuous group homomorphism from G to the multiplicative group of positive real numbers .
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 locally compact and such groups have a natural measure called the Haar measure.
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure.Using the Haar measure, one can define a convolution operation on the space C c (G) of complex-valued continuous functions on G with compact support; C c (G) can then be given any of various norms and the completion will be a group algebra.
Since every operator in SU(2) is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G. The Lebesgue measure on the real line is a special case of this.
The representations of the group are found by considering representations of (), the Lie algebra of SU(2).Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; [1] we will give an explicit construction of the representations at the group level below.
The representative functions A form a non-commutative algebra under convolution with respect to Haar measure μ. The analogue for a finite subgroup of Γ of SU(2) is the finite-dimensional group algebra C[Γ] From the Clebsch-Gordan rules, the convolution algebra A is isomorphic to a direct sum of n × n matrices, with n = 2j + 1 and j ≥ 0.
The Haar measure is an invariant mean (unique taking total measure 1). The group of integers is amenable (a sequence of intervals of length tending to infinity is a Følner sequence). The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn–Banach theorem this way.