$m$维紧区间上变分McShane积分的完全刻画
Full characterizations of the variational McShane Integral on $m$-dimensional compact intervals
摘要：
本文考虑定义在$m$维欧氏空间$\mathbb{R}^{m}$中三次区间$C_{m} = [0,1]^{m}$的所有非退化闭子区间族$\mathcal{I}_{m}$上的可加区间函数，其取值于Banach空间$X$ .利用$F$的凸三次平均值域给出了可加区间函数$F : \mathcal{I}_{m} \to X$是变分McShane（或强McShane）可积函数$f : C_{m} \to X$的本原的充要条件.
英文摘要：
In this paper we consider the additive interval functions defined on the family $\mathcal{I}_{m}$ of all non-degenerate closed subintervals of the cubic interval $C_{m} = [0,1]^{m}$ in the $m$-dimensional Euclidean space $\mathbb{R}^{m}$ and taking values in a Banach space $X$. We give necessary and sufficient conditions for an additive interval function $F : \mathcal{I}_{m} \to X$ to be the primitive of a variational McShane (or strong McShane) integrable function $f : C_{m} \to X$ in terms of the convex cubic average range of $F$.
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