316 MARCUS AND MIZEL 5. ~-INVARIANT FUNCTIONALS; PROOF OF LEMMA 4.5 PROPOSITION 5.1 [9; Sect. 41. Let A’ be a rich subspace <strong>of</strong> L=(m) and let X be an odd, disjointly additive functional on .y,[ If f, g, h are disjoint simple functions such that f - g and f - h belong to Y/J , then iv(f - g) = - N(f - I/) - X(g - I?). (5.1) The same statement holds if YJ. is replaced by
EXTENSION THEOREMS FOR FUNCTIONALS 317 Next we claim that, <strong>for</strong> every triple <strong>of</strong> positive numbers, a, 6, C, Since the equality is symmetric with respect to a, b, c, we may and shall assume that c 3.: mas(a, b). Given a set S in 7, let (S, , S,} be a partition <strong>of</strong> S into sets <strong>of</strong> equal p-measure. Further, let (Sr’, $1 be an (a, h) partition <strong>of</strong> S, and let S,’ be a subset <strong>of</strong> S, such that By (5.5) p(&‘) = (ub/c(a 4- h)) /L(S$). Ya,b(w r= ; Y,,,(S), These equalities, together with (5.2), yield, For the last two equalities we used the fact that &’ is odd in conjunction with Proposition 5.1. Now let c,, be a fixed positive number, and set By (5.6) and (5.7), yn = Ul%) Yo.c,, 1 vu T- 0. (5.7) YdS) = @a.(S) - qqq (5.8) Let h E fFi, , h = uxu - 6xr, ) U n If :-= c‘(, n, h positive. By (5.2) (5.5), (5.7), and (5.8), we have, The measure rU is absolutely continuous with respect to nz. Hence, by the Radon-Kikodym theorem, p,, possesses a density function in Ll(m) which we
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- Page 47 and 48: such that and Pvoqf. We denote On C
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A DlJNFORD-PETTIS THEOREM FOR L1/Hu
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the mapping A DVNFORD-PETTIS THEORE
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A DCNFORD-PETTIS THEOREM FOR LIIHu
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.t DCNFORD-PETTIS THEOREM FOR L’,
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A DUNFORD-PETTIS THEOREM FOR Ll/H=-
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Lebesgue’s theorem implies that A
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,omivnr. OF FLW~~IONAI. ANALYSIS 24
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REPRESENTATIONS OF NILPOTENT LIE GR
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IIEPRESENTATIOKS OF NILPOTENT LIE G
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I
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REPRESENTATIONS OF NILPOTENT LIE GR
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REPRESENTATIONS OF SII.PoTEST LIE C
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REPRESENTATIONS OF NILPOTENT LIE GR
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KWKESE~TATIOSS OF NILPOTENT LIE (:I
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REPRESENTATIONS OF NILPOTENT LIE GR
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JOURNAL OF FUNCTIONAL ANALYSIS 24,
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TOPOLOGI('AI. ASPECTS 399 maps i,,
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where TOPOLOGICAL ASPECTS 401 1,’
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TOPOLOGICAL ASPECTS 403 PROPOSITION
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TOPOLOGICAL ASPECTS 405 (iv) S is b
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TOPOLOGICAL ASPECTS 407 ever!- o-co
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TOPOLOGICAL ASPECTS 409 I,et a = (J
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The polar decomposition of the map
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TOPOLOGICAL ASPECTS 413 Proof. It i
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TOPOLOGICAL ASPECTS 415 algebras a
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TOPOLOGICAL ASPECTS 417 where M, <
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‘I’OPOLOCICAL ASPECTS 419 RIore
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By the same method: TOPOLOGICAL ASP
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TOPOLOGICAL ASPECTS 423 Since f bel
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TOPOLOGICAL ASPECTS 425 20. H. H. S
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