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=∑k=1nwk→∞ 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 σn(x) is given byσn(x)=1Wn∑k=1nwkxk,n=1,2,….If in addition, E(ℳ) has absolutely continuous norm, then, (σn(x))n ≥ 1 converges in E(ℳ) if and only if x =(xn)n ≥ 1 is uniformly integrable and its limit in measure topology x∞ ∈ E(ℳ).