中文摘要：

设P 是一个概率测度, ψ 是一个复值可积函数, dμ =ψdP是一个复值测度. 在权函数ψ∈a1∩b∝＋ 和Banach空间X 具有适当的凸性和光滑性的条件下, 作者证明了关于复测度μ 的X 值拟鞅空间Dα（X） 和pQα（X） 上的原子分解定理. 并且利用复测度拟鞅的原子分解定理, 在0〈α ≤ 1 的情形, 证明了关于X 值复测度拟鞅的两个重要不等式.

英文摘要：

Let P be a probability measure, ψ a complex valued integrable function and dμ= ψdP a complex valued measure. Two theorems of atomic decompositions for the space Dα（X） and pQα（X） of X-valued quasi-martingales with respect to the complex measure μ are obtained when ψ∈a1∩b∝＋ and the Banach space X has suitable convexity and smoothness. As the applications of atomic decompositions two inequalities are proved for X-valued quasimartingales respect to the complex measures μ by using atomic decompositions in the case of 0〈α〈1.