Double integral theorem of Haar measures

Markov-Kakutani Theorem and Haar Measure

Conditional Haar Measures on Classical Compact Groups

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads to the following result: for this conditional measure, writing Z (p) U for the first nonzero derivative of the characteristic polyn...

Cauchy Integral Theorem

where we use the notation dxI for (1.4) dxI = dxi1 ∧ dxi2 ∧ ... ∧ dxik for I = {i1, i2, ..., ik} with i1 < i2 < ... < ik. So ΩX is a free module over C ∞(X) generated by dxI . Obviously, Ω k X = 0 for k > n and ⊕ΩX is a graded ring (noncommutative without multiplicative identity) with multiplication defined by the wedge product (1.5) ∧ : (ω1, ω2)→ ω1 ∧ ω2. Note that (1.6) ω1 ∧ ω2 = (−1)12ω2 ∧ ω...

Egoroff Theorem for Operator-Valued Measures in Locally Convex Cones

In this paper, we define the almost uniform convergence and the almost everywhere convergence for cone-valued functions with respect to an operator valued measure. We prove the Egoroff theorem for Pvalued functions and operator valued measure &theta; : R &rarr; L(P, Q), where R is a &sigma;-ring of subsets of X&ne; &empty;, (P, V) is a quasi-full locally convex cone and (Q, W) is a locally ...

Double Integral Characterization for Bergman Spaces

&lrm;In this paper we characterize Bergman spaces with&lrm; &lrm;respect to double integral of the functions $|f(z)&lrm; &lrm;-f(w)|/|z-w|$,&lrm; &lrm;$|f(z)&lrm; -&lrm;f(w)|/rho(z,w)$ and $|f(z)&lrm; &lrm;-f(w)|/beta(z,w)$,&lrm; &lrm;where $rho$ and $beta$ are the pseudo-hyperbolic and hyperbolic metrics&lrm;. &lrm;We prove some necessary and sufficient conditions that implies a function to be...