Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

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320 MARCW AND MIZIX Then the following assertions hold. Without loss of generality, we may and shall assume that urr :. I for all rr. By Proposition 3.2, there exists a function /I,,* of the form such that Sn*, Z’,,” are disjoint subsets of B,, and a,,(~~ . II XT.,,*) By Proposition 3.5(a), the measure (m, p) : 7 + K )I l1 is a Lyapunov measure. Hence, there exist subsets S,, , I”,, of Sn*, T,,“, respectively, such that (7% P)(S,J ~~~~ 44 I*)(&,“‘), (w cL)( 7’4 a,,(m, p)(T,,"), 77 I, 2.... Set II,, xr -~~ xs, . Then, by (6.7) and (6.8), . I/ (6.7) (6.X) Nest, let ;\I be a bound for the sequence [.f,,l. Again, by Proposition 3.2, there exists a function g, of the form 2,16(,yCT,, - x,,,~) such that U, , I Vri are disjoint subsets of A,( and Set, p,, f/n sn (n I, 2 ,... ). Then, by (6.6) (6.7), (6.9). and (6.10) CT,! is a bounded sequence in L”(m) such that From (6.11) it follows that {P)~} is (bm)-convergent to zero. ‘Ihus {I;, ~~ v,!I is (bm)-convergent to ,f. Further, by (6.1 I), (fn - v,,)
EXTENSION THEORERIS FOR FUNCTIONALS 321 But H(fn ~~ 4(t) H(fn)(t) for all t in Q:,K(~I,J. Therefore, (6.12) and (6.13) imply (6.3). LEMMA 6.3. Let {fill be a (bm)-convergent sequence of simple functions. Then {H(f,,)\ conz!erges in IdI( I’Yocf. Let f he the (bm)-limit of if,,:. First vvc consider the cast where ,f is in A. Denote ‘Then, where Clearly, Thus, b\ 1i.j I’ I H(.fi) - H(fj)l dft2 [ (H(xT,j) -- H(l2j.j)) C/II?, (6.14) 3’ i2 - R Lemma 6.2, (bnl)-Jim gi,j : (bm)-lip h,,i mm f. Z,J"" 1,J~ )Z By (6.14) and (6.15), limi,j,7j Z,,j 0. Nest we consider the general case where f is not necessarily in A!. By Proposi- tion 3.5(b), the indefinite integral off with respect to p is a Lyapunov measure. Hence, there exists a partition of Q, say (9, , Q,}, such that, Set f” = f(xr), -- x~I,) and fiL” = fn(xal - xs+ a =--~ 1, 2,.. . . Then f-% E . c’/ and :f,l”‘) is (bm)-convergent to f *. Thus, by the first part of the proof, {H(f?! “)I converges in Ll(m). Since H(f,)xo, = H(f?z*)xn, this implies that (H(f&J converges in Lr(m). Similarly, one shows that [H(fJxa2} converges in Z,‘(m). This completes the proof of the lemma. ~XhIMA 6.4. The mapping H: YA L’(m), defined by (6.1), possesses nn extension 2: L”(m) ---f L’(m) which is disjointly additive and continuous ,with respect to the (bm)-topology of L7-(m) and the norm topology of Ll(nz). Furtllermore, we have fl(fXE) S(f)XE 7 VftL-f(m), VEE 7.
Page 1 and 2: Extension Theorems of Hahn-Banach T
Page 3 and 4: EXTENSIOK THEOREMS FOR FUNCTIOK.4LS
Page 5 and 6: EXTENSION THEOREMS FOR FL'NCTIONALS
Page 7 and 8: EXTENSION THEOREMS FOR FLJNCTIONALS
Page 9 and 10: EXTEIiSION THEOREMS FOR FUSCTIONALS
Page 11 and 12: EXTENSION THEOREMS FOR FUNCTIONALS
Page 13 and 14: Hence, by Proposition 4.6, EXTENSIO
Page 17: ESTENSION THEOREMS FOR FIVVCTIONALS
Page 21 and 22: EXTENSION TfIEOREMS FOR FUNCTIONALS
Page 23 and 24: EXTENSION THEOREMS FOR FUNCTIONAW 3
Page 27 and 28: EXTENSION THEOREMS FOR FUNC’l’I
Page 29 and 30: l\jow, let f be a function in F,/ ,
Page 33 and 34: KXTEXSION THEOREMS FOR FLWCTIOXA1.S
Page 35 and 36: MEASURE SPACE AI-TOiVIORPHIBRIS 337
Page 37 and 38: YIEASURE SPACE AUTOMORPIIISMS 339 W
Page 39 and 40: C‘hoosc a representative (F,,),,N
Page 41 and 42: MEASURE SPACY AUTOMORPIIIS~I~ 343 (
Page 43 and 44: MEASURE SPACE AYI’O~IORPHIS~ld 34
Page 45 and 46: MEASURE SPACE AUTOMORPHISMS 347 (3.
Page 47 and 48: such that and Pvoqf. We denote On C
Page 49 and 50: 4. ~WPLICATIONS \\‘c want to desc
Page 51 and 52: The Theta Transform and the Heisenb
Page 53 and 54: THE THETA TRANSFORRZ 355 The origin
Page 55 and 56: THE THETA TRANSFORM 357 It is conve
Page 57 and 58: THE THETA TRAXSFORM 359 zohew x1 T
Page 59 and 60: TIIE TfIEI‘A ‘lxANSFOK~1 361 Fo
Page 61 and 62: THE THETA TRANSFORhI 363 where J u,
Page 63 and 64: A DUNFORD-PETTIS THEOREM FOR L1/Hal
Page 65 and 66: A DlJNFORD-PETTIS THEOREM FOR L1/Hu
Page 67 and 68: 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
Page 79 and 80:
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 SII.PoTEST LIE C
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KWKESE~TATIOSS OF NILPOTENT LIE (:I
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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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Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

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