For X a compact abelian group and B an infinite subset of its dual X̂, let CB be the set of all x ∈ X such that 〈φ(x) : φ ∈ B〉 converges to 1. If F is a free filter on X̂, let DF = {CB : B ∈ F}. The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F . We show that there is a filter F such that DF has measure 0 but is not contained in any CB . Thi...

This is a set of lecture notes which present an economical development of measure theory and integration in locally compact Hausdorff spaces. We have tried to illuminate the more difficult parts of the subject. The Riesz-Markov theorem is established in a form convenient for applications in modern analysis, including Haar measure on locally compact groups or weights on C∗-algebras...though appl...

The operator A(D) is called the radial component of D. Many special cases have been considered (see e.g. [1, §2], [4, §5], [5, §3], [7, §7 ], [8, Chapter IV, §§3-5]). Suppose now dv (resp. dw) is a positive measure on V (resp. W) which on any coordinate neighborhood is a nonzero multiple of the Lebesgue measure. Assume dg is a bi-invariant Haar measure on G. Given u E Cc (G X W) there exists [7...

This paper introduces a generalization of Pontryagin duality for locally compact Hausdorff abelian groups to locally compact Hausdorff abelian group bundles. First recall that a group bundle is just a groupoid where the range and source maps coincide. An abelian group bundle is a bundle where each fibre is an abelian group. When working with a group bundle G we will use X to denote the unit spa...

In [6], M. Krishnapur and the authors considered the convergence of the empricial measure of (complex) eigenvalues of matrices of the form An = TnUn, where Un is Haar distributed on U(n), the unitary group of n×n matrices, and independent of the self-adjoint matrix Tn (which therefore can be assumed diagonal, with real non-negative entries s i ). That is, with λ (n) i denoting the eigenvalues o...

A compact Lie group G and a faithful complex representation V determine the Sato-Tate measure μG,V on C, defined as the direct image of Haar measure with respect to the character map g 7→ tr(g|V ). We give a necessary and sufficient condition for a Sato-Tate measure to be an isolated point in the set of all Sato-Tate measures, regarded as a subset of the space of distributions on C. In particul...

We classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple S-algebraic groups invariant under a maximal split torus in at least one simple local factor, and show that the algebraic support of such a measure splits into the product of four homogeneous spaces: a torus, a homogeneous space on which the measure is (up to finite index) the Haar measure, ...

We obtain an optimal deviation from the mean upper bound D(x) def = sup f∈F μ{f −Eμf ≥ x}, for x ∈ R (0.1) where F is the complete class of integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of (0.1) for Euclidean unit sphere Sn−1 with a geodesic distance function and a normalized Haar measure, for R equipped with a Gaussian measure and...

For X a compact abelian group and B an infinite subset of its dual X̂, let CB be the set of all x ∈ X such that 〈φ(x) : φ ∈ B〉 converges to 1. If F is a free filter on X̂, let DF = {CB : B ∈ F}. The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F . We show that there is a filter F such that DF has measure 0 but is not contained in any CB . Thi...