无RNP性Banach格中鞅型序列的收敛性

CONVERGENCE OF MARTINGALE-LIKE SEQUENCES IN BANACH LATTICES WITHOUT （RNP）

摘要: 本文证明了（1）设E是序连续Banach格,\left(x_n, \mathscrF_n\right)_n>1是满足条件（C）的subpramart,若存在a.e.强收敛的强可测函数列（y_n）_n≥1,使有 0≤x_n≤y_n,\forall n \geqslant 1,,则（x_n）_n≥1a.e.强收敛.且每一个 E⁺值反向subpramart a.e.强收敛。（2）设E是AL空间,若\left(x_n, \mathscrF_n\right)_n>1是E⁺值superpramart,则TLx_na.e.存在。

Abstract: In this paper it is proved that （1） under the supposition E is an order continuous Banach lattice and\left(x_n, \mathscrF_n\right)_n>1 is a subpramart of class （C）, if there is an a. e. strongly convergent and strongly measurable sequence （y_n）_n≥1 such that 0≤x_n≤y_n, n≥1, then （x_n）_n≥1 is a. e. strongly convergent, and every E⁺-valued reversed subpramart is a. e. strongly convergent; （2） under the supposition E is an AL space, if \left(x_n, \mathscrF_n\right)_n>1 is an E⁺- valued superpramart, then TL x_n a. e. exists.