Isotonic approximation in L1

[摘要] Let (Omega,A,P) be a measurable space and L subset of A a sub-sigma-lattice of the sigma-algebra A. For X is an element of L-1 (Omega,A,P) we denote by PLX the set of conditional 1-mean (or best approximants) of X given L-1(L) (the set of all L-measurable and integrable functions). In this paper, we obtain characterizations of the elements in PLX, similar to those obtained by Landers and Rogge for conditional s-means with 1 < s < infinity. Moreover, using these characterizations we can extend the operator P-L to a bigger space L-0(Omega,A,P). When, in certain sense, L-n goes to L-infinity, we will be able to prove theorems about convergence and we will obtain bounds for the maximal function \sup(n) PLnX\. A sharper characterization of conditional 1-means for certain particular sigma-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered sigma-lattices. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-08-01

[关键词] best approximants;maximal inequalities;a.e. convergence