By Charles Arthur Conant
...With An Account Of the industrial Crises Of The 19th Century And The concern Of 1907
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E/. E/-valued Pettis-integrable multifunctions satisfying: (i) E Fq jXn j < 1 for each n 2 N and each 1 Ä q < n. E/-valued Pettis-integrable martingale, that is, P e-E Fn XnC1 D XRn for all n 2 N. x ; Xn /jdP < 1. Am /. E/-tight for each m 2 N. s. to X1 . Proof. F /. i i / exists. m/ . Am /. m/ 30 F. Akhiat et al. s. Am /. x ; X1 / D hm x a:s: ! x ; X1 / a:s: ! x; X1 / a:s: ! m/ P1 m Put X1 D mD1 X1 1Am . s. to X1 . It is easy to check that X1 is F -measurable. s. to a function in L1 . c0 6,! E/ any F -measurable and scalarly integrable selection of X1 is Pettis-integrable.
Am /. x ; X1 / D hm x a:s: ! x ; X1 / a:s: ! x; X1 / a:s: ! m/ P1 m Put X1 D mD1 X1 1Am . s. to X1 . It is easy to check that X1 is F -measurable. s. to a function in L1 . c0 6,! E/ any F -measurable and scalarly integrable selection of X1 is Pettis-integrable. Accordingly X1 is Pettis-integrable. 2. c0 6,! E/. Xn ; Fn /n2N be an adapted sequence of E-valued Pettis-integrable functions satisfying: (i) E Fq jXn j < 1 for each n 2 N and each 1 Ä q < n. Xn ; Fn /n2N is an E-valued Pettis-integrable martingale, that is, Xn D P e-E Fn XnC1 for all R n 2 N.
His core equivalence theorem is very general in that he assumed a set of very weak conditions on traders’ preferences. In fact, in his We are grateful to Prof. Ezra Einy for his valuable comments and pointing out some related references, and would like to thank an anonymous referee and the editor for their helpful comments. S. Kusuoka, T. 1007/978-4-431-99490-9 2, c Springer 2010 35 36 J. -I. Takekuma proof, neither irreflexivity nor transitivity is assumed on traders’ preference relations. Our purpose is to revisit his equivalence theorem under somewhat different or general assumptions.
A History Of Modern Banks Of Issue by Charles Arthur Conant