Convergence of Nonmonotone Sequence of Sub- -Fields and Convergence of Associated Subspaces LpBn‘ p 2 ’1; C1“‘

Kud¯o [Ann. Probab. 2(1) (1974), 76–83] introduced a notion of convergence of sub- -fields in terms of L1-convergence of conditional expectation. We study relationships between convergence of sub- -fields and convergence of associated subspaces LpBn‘. In particular, we show that the Slice convergence is not adapted to the L1-case in contrast to the Mosco convergence. Notions of upper and lower limits are also studied. This enables us to obtain measurability results regarding those limits. We also give examples and counterexamples in order to exhibit characteristic behaviours of the sequences Bn‘n, LpBn‘‘n, and  Bn ‘n.