CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x =(xn) n ≥1 be a martingale on a noncommutative probability space(M,τ) and(wn) n ≥1 a sequence of positive numbers such that Wn = ∑nk=1 wk →∞ as n →∞.We prove that x =(xn) n≥1 converges in E(M) if and only if(σn(x))n≥1 converges in E(M),where E(M) is a noncommutative rearrangement invariant Banach function space with the Fatou property and σ n(x) is given by k=1 If in addition,E(M) has absolutely continuous norm,then,(σ n(x)) n ≥1 converges in E(M) if and only if x =(x n) n ≥1 is uniformly integrable and its limit in measure topology x ∞∈ E(M).