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In 1936, André Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure with a certain separating property, [3] 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 the Haar measure on this completion.
Any compact group is locally compact.. In particular the circle group T of complex numbers of unit modulus under multiplication is compact, and therefore locally compact. The circle group historically served as the first topologically nontrivial group to also have the property of local compactness, and as such motivated the search for the more general theory, presented here.
Compact groups all carry a Haar measure, [6] which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to positive reals (R +, ×), and so 1). In other words, these groups are unimodular. Haar measure is easily normalized to be a probability measure, analogous to dθ/2π on the circle.
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.
which would say the left translate of a right Haar measure is a multiple of a left Haar measure. However, the left translate of a right Haar measure is also a right Haar measure so what you're looking for is an assertion that a right Haar measure is a multiple of a left Haar measure, e.g. is itself left Haar. That's only true for unimodular groups.
A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n − 1. A maximal torus in the special orthogonal group SO(2 n ) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces).
The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed. Spherical measure has a nice relationship to Haar measure on the orthogonal group. Let O(n) denote the orthogonal group acting on R n and let θ n denote its normalized Haar measure (so that θ n (O(n)) = 1
Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., () = for every Borel measurable set, where is the Borel measure described above). This idea extends to finite-dimensional spaces R n {\displaystyle \mathbb {R} ^{n}} (the Cramér–Wold theorem , below) but does not hold, in general, for infinite-dimensional spaces.