# 4-Dimensional projective planes of Lenz type III - download pdf or read online

By Salzmann H.

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One direction, namely ,1(K u L) ~ ,1(K) + ,1(L) is obvious, so it remains to be shown that for arbitrary ,1(K) + ,1(L) ~ ,1(K u L) B > 0 + B. By definition there is an open set WEd containing K u L such that A(W) - ,1(K u L) < B. e. we have K £G, L £H, Gn H = 0. Hence ,1(K) + ,1(L) ~ A(G n W) + A(H n W) = A«G u H) n W) ~ thus finishing the proof. A(W) < ,1(K u L) + B, D 24 2. Radon Measures and Integral Representations Later on we shall need existence and unicity of certain Radon measures on the product of two Hausdorff spaces X and Y not only for the product of two measures, but also for so-called Radon bimeasures.

In particular both limits exist and agree with respect to the weak topology. 12. Exercise. Let Sand T be two Hausdorff topological semigroups and let h: S ~ T be a continuous homomorphism. Then for Ji, v E M~(S) we have §4. Vague Convergence of Radon Measures on Locally Compact Spaces In this section X denotes a locally compact Hausdorff space. 1. Definition. e. the coarsest topology in which the mappings /l ~ (Ji,j> are continuous, when j ranges over CC(X), cf. 10. In particular the vague topology is a Hausdorff topology.

For every positive linear functional L: V ~ IR there exists a Radon measure J1 on X such that V £; ,pl(J1) and PROOF. L(f) = ff {f C(X)llfl ~ g for some g E V+}. f E V. dp We define v= E Then V is a subspace of C(X) containing V, and a simple compactness argument combined with (i) shows that CC(X) £; V. We claim that V= V+ + Jt: In fact, iff E Ii and g E V+ is such that If I ~ g, thenf = (g + f) + (-g) shows the assertion. 7 it follows that L can be extended to a positive linear functional L: V ~ IR, and by the Riesz representation theorem there exists a Radon measure J1 on X representing [ ICC(X).

### 4-Dimensional projective planes of Lenz type III by Salzmann H.

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