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By A. Gut, K. D. Schmidt

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Vector of the of set f u n c t i o n s for all of processes. process for amarts further a wide amarts process This a p p r o a c h been d e v e l o p e d B e l l o w and E g g h e continuous) . This p r o p e r t y D n ( B n v ~n ) interesting property type. e. of transparent. 3, to a processes in w h i c h , of operator derivative also b e c o m e additive operator direct This c o n s t r u c t i o n derivative, on the w h o l e who r e d u c e d the Y o s i d a - H e w i t t a more from w h i c h derivative each b o u n d e d generalized By integration, arguments • LI(~, ~) Radon-Nikodym As an example, Radon-Nikodym Radon-Nikodym homomorphism present that the g e n e r a l i z e d ba(F, ~) which associates F we shall Radon-Nikodym [35,36] case by u s i n g is not needed.

Stability This section deals with the following problem: Given an smart ~: R ~ and a function R, {~(Xn)'~n}nED when is an ~m~rt? 6). For the case D •N, and sufficient conditions on Here the case D =-N {Xn}nCD is Ll-bounded when D=N Bellow (1976 b, 1977 ) gives necessary ~ for the conclusion to hold in general. will also be covered. The proofs differ slightly from those given in Bellow (1977) . We also investigate which further assumptions on the amart one needs for the conclusion to remain valid when the necessary conditions on ~ no longer are satisfied.

2g n Now, let where 0 < r < I. =I z% L Z -n --E(i~iIrISrn) ' n = 1,2, .... I k=l n-l. , -3, -2, -I . 8) implies that T-uniformly integrable. 10. 16. 2 for the case i < r < 2. n_i/r, n This was possible because E sup I E ~kI < ~ . 2 is not applicable because E sup n -I. 2) and the latter condition was not assumed. 2 does not. 17. 15 can be used to illustrate the importance of the a-algebras relative to which a sequence is adapted. Consider the case 0 < r < i D -- -N and suppose further that and ~-n = ~ { ~ ; k <__-n} ~i' ~2"'" all are non-negative.

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