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a e G A-element a.e. x e G abelian group absolutely continuous analytic and harmonic analytic functions archimedean Banach space Borel function Borel measure Borel sets Borel subset boundary values Cauchy sequence Chapter compact Hausdorff space completes the proof continuous function convergence Corollary cp-analytic defined on G Definition denote direction of cp dual groups f is harmonic f|dA f|Pdm Fourier-Stieltjes transform fr(x Fubini theorem function f function on G functions defined group G groups whose dual h(x+y)du Haar measure half-plane Hardy spaces harmonic functions harmonic polynomials Hausdorff space hence homomorphism isometrically isomorphic lim M f](r linearly ordered LP(A LP(G mapping measure on G non-negative p e Hom(S,E p-integrable Proposition prove quasi-harmonic functions real numbers regular Borel measure representing function resp respect to Haar satisfies semigroup of type shows Space HP(A subspace topology u e M(G unique unit disk