Download e-book for kindle: A crash course on Kleinian groups; lectures given at a by American Mathematical Society

By American Mathematical Society

ISBN-10: 0387068406

ISBN-13: 9780387068404

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Additional info for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

Sample text

This algebra contains, of course, the products of open sets in X (resp. Y) and therefore a base for the topology on Z. l( Y). Let us now first assume that <1>(X, Y) < 00. 91, G open} for each compact set C £ Z. 91 and K(K x L) ~ A(K x L) = <1>(K, L) by monotonicity of A. On the other hand, we may use the two finite Radon measures ,u(A):= <1>(A, Y) on X and v(B):= <1>(X, B) on Y to provide us 25 § 1. 1(G\K) 8. e. we have the desired equality. If A E 81(X), B E 81(Y) and C is a compact subset of Ax B, then the projections K:= nx(C) and L:= ny(C) are still compact and C £ K x L £ A x B, implying K(A x B) = sup{K(C)1 C £ A x B, C E ff(Z)} = sup{K(K x L)IK £ A, L £ B, K E ff(X), L = sup{(K, L)IK £ A, L £ B,KE ff(X), L E E ff(Y)} ff(Y)} = (A, B), using in the last equality once more that is a Radon bimeasure.

F3 E A, BE 84(X)), and let the set function fl: 84(X) fl(B) = sup flex(B) = lim flex(B) exEA exEA f3 => flex(B) ~ flp(B) for all [0, ooJ be defined by rJ. ~ ~ for B E 84(X). 33 §2. The Riesz Representation Theorem Show that if Ji(K) < measure on X and If 00 dJi for all compact sets K = sup (lEA If dJi(l If = lim (lEA for all Borel measurable functions f: X --. [0, £; 00 X then Ji is a Radon dJi(l J. Furthermore, for all Ji-integrable functions f: X --. C. 30. Exercise. e. all sets {n} £; ~ are open and a subset G £; X containing 00 is open if and only if its "density" lim n _ oo (ljn)IGn {1, ...

I/n} ni=l ~ f we get lim inf f f d~a ~ ~ ~ lim inf ~a({f > ~}) ~ ~ ~ ~({f > ~}). J where the last expression converges to f dJ1 as n tends to 00. The implication "(iv) => (ii)" is obvious. If f: X ~ lR is bounded and continuous and J1(l ~ J1 weakly, then by (iv) and (v) f dJ1(l ~ f dJ1. Now suppose that X is completely regular and lim f dJ1(l = f dJ1 for all continuous bounded f: X ~ IR. To show (ii) let G ~ X be open and let K £; G be compact. As an immediate consequence of the very definition of complete regularity we fi~d a continuous function f: X ~ [0,1] such thatflK == 1 andflG c == o.

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A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco by American Mathematical Society


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