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0 - measure 1. Thus we can take Q to be Xr' R. This is in many ways more natural. By adding the oo's above, we get a "cheap" proof of the countable additivity of dµ. 0 (only "cheap" by relying on functional analytic machinery). , Gel'fand-Vilenkin [54], Glimm-Jaffe [71]. 2. 9. For any (real) Hilbert space, J{, the Gaussian process indexed by J{ exists. 3. The above model for Q is, of course, basis dependent. , Section I. 6) and a sin (it's "unnatural"). Model 2 (Spectrum Model). 6. 's associated to two different choices of Q.

Sc (or c- 2 ::;µn:Sc 2). 52), 1\ · 1\p is an LP(Q,dµ 0 ) norm. 53a) where (1. 53b) f) Pick p = 2/1- (2c)- 1 so that 2 = p(1-(~- 1 > p 1- ~> p(l-µ), so the integral converges. 53) is clearly C 00 for µ in (c- 2 , c 2 ) and g(l) = 1; g'(l) = 0. 54) where T/ = (4pr 1 sup g ... (µ) . 54). 5. 48), and c- 1 < ,\n < c I(µn- 1) 2 n = I

Proof. 26) on the product space QJ{ x QJ{ 1 with the 2 product measure. v. are jointly Gaussian 2 and thus their sum is a g. r. v. 8. Let K be a (closed) subspace of J{, a real Hilbert space. Then there is a natural imbedding, H, of rcK) into r(}{) so that (1) H¢K(f) = ¢J{(f)H for aII f € K (2) a : :JRK ... :JR}{ defined by a(F(¢K(fi))) = F(¢J{(fi)) defines a homomorphic imbedding of :JRK into :JR}{; in particular a takes positive (boundecf) random variables on QK into positive r. v. on QJ{" Proof.

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Advansed plasma theory by Rosenbluth

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