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# Séminaire de Probabilités XXIV 1988/89 by Michel Ledoux (auth.), Jacques Azéma, Marc Yor, Paul André

By Michel Ledoux (auth.), Jacques Azéma, Marc Yor, Paul André Meyer (eds.)

The diverse papers contained during this quantity are all study papers. the most instructions of study that are being built are: quantum likelihood, semimartingales and stochastic calculus.

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18) converges to 0 in L2(wr(x)dx) norm. Proof of Theorem Remarks. 1. 1. 1. We have shown t h a t T is a b o u n d e d o p e r a t o r on [] LP(dx) with a bound independent of the L 1 n o r m of T. So one could dispense with the hypothesis t h a t T is integrable b y a suitable limiting process (cf. [17]). 2. O p e r a t o r s such as the t r u n c a t e d Hilbert t r a n s f o r m T* ([17], p. 38) can b e written as a s u m T~ -4- T~, where T~ satisfies the hypotheses of T h . l . 1 with constants cl and c2 independent of ¢ a n d T~ is integrable with L 1 n o r m independent of ¢.

Z.. ,>. ),z,. z =I16 + I17 + h8 + I19. 27) is bounded by c2 -k/2. When [zl > 2 k/:, r~p(z)/wr(z) <_ clzl ~-~ < c2 -k~. 27) is also bounded by c2-k/~; hence so is I16. 16), I17 '<~Cf < f p(Z)Wr(Z)--l[19k(y Z) -[- 19k(z)]dzwr(x)dx - - I>:l J c,"<' f K~)w~(z) -l[pk(~ - z) + p r ( z ) ] d z . 27). 28) Ils < c i f p(z- x)wr(z)-l[pk(y-- z) + pk(z)]w(x)dx dz A il An(l~:l>lzlt~) is, Ar,(P:I 1, Iz - x I > 2k/s). 28) is bounded by If Ixl -< Izl/2, then p(z - x) < ~p(=), Izl > 2 > ,-, ,~nd Izl > c2 ~/8.

5) (appliqu6e en prenant F = f ( E ) ) que h est mesurable en rant qu'application du sous-espace f ( E ) de F dans l'espace G. Puisque celui-ci est injectif, h peut 6tre prolong6e en une application mesurable de l'espace F tout entier darts G, et cela prouve que G poss~de la propridt6 de Doob. 2) P r o p o s i t i o n . Les espaces de Doob s@arables sont les espaces mesurables lusi. niens( au sens de [1]). Par consdquent, tous les espaces de Doob sdparables non ddnombrables sont isomorphes (en particulier, isomorphes h R ou h {0, 1}N).