A-收敛与几乎处处收敛

设A≡(a_i)^∞_i=1?S+_<sub>l₁,其中,S</sup>+_<sub>l₁表示l₁单位球面上的所有正向量构成的集合.Banach空间X中的序列(x_n)称为A-收敛于x∈X,是指对任意的ε>0,lim_i→∞〈a_i,χ_A(ε)〉=0,其中,A(ε)={n∈N∶‖x_n-x‖≥ε}.用两种不同的收敛方式刻画A-收敛,即证明对任意A≡(a_i)∞_i=1?S</sup>+_<sub>l₁,存在一个N上的理想I_A,以及一族极端有限可加概率测度P_ext(I_A),使A-收敛且理想I_A-收敛和测度P_ext(I_A)-收敛互为等价.此外,证明A-收敛为测度P_ext(I_A)-几乎处处收敛的充分必要条件是该A-收敛为非退化的.</sup>

On A-Convergence and Almost Usual Convergence

Let A≡(a_i)^∞_i=1?S⁺_l1, a sequence(x_n)of points in a Banach X is said to be A-convergent to x∈X provided that for any ε>0, lim_i→∞〈a_i,χ_A(ε)〉=0,where A(ε)={n∈N∶‖x_n-x‖≥ε}. In this paper, we give two different approaches of A-convergence via ideal on N and via extreme measures. We show that for any A≡(a_i)^∞_i=1?S⁺_l1, there exist an ideal I_A and a collection P_ext(I_A)of extreme probability measures such that the A-convergence, the ideal I_A-convergence and the measure P_ext(I_A)-convergence are equivalent. We also show that A-convergence equivalent to P_ext(I_A)-almost usual convergence if and only if it is nondegenerate.