On interval based generalizations of absolute continuity for functions on $\mathbb{R}^n$
Michael Dymond, Beata Randrianantoanina, Huaqiang Xu
备注：
29 pages, 2 figures, this is a new version which includes a new co-author and new results, one of the results from the previous version was separated into a different paper, since it is of independent interest and to reduce the length of the paper (which even now is 29 pages long)
We study notions of absolute continuity for functions defined on $\mathbb{R}^n$similar to the notion of $α$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish containment relations of the class $1-AC_{\rm WDN}$ which consits of all functions in $1-AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.
Shape Analysis, Lebesgue Integration and Absolute Continuity Connections
由于Srivastava和Klassen的教科书“Functional and Shape Data Analysis”中提出的形状分析与Lebesgue积分和绝对连续性有着错综复杂的关系，因此很好地掌握后两个概念是有利的。因此，在这些笔记中，我们审查的基本概念和结果的勒贝格积分和绝对连续性。特别是，我们审查的基本结果连接到彼此和形状分析的类型，或更一般地说，功能数据分析在上述教科书中提出的，在这个过程中阐明所有三个概念的重要方面。许多著名的结果，特别是大多数关于Lebesgue积分的结果和一些关于绝对连续性的结果，在没有证明的情况下被给出。然而，相当数量的结果绝对连续性和大多数结果的功能数据和形状分析的证明。事实上，大多数失踪的证明可以发现罗伊登的“实分析”和鲁丁的“数学分析原理”，因为它是对这些经典教科书和斯利瓦斯塔瓦和克拉森的教科书，很好的一部分，这些笔记的基础。然而，如果一个结果的证明没有出现在上述教科书中，也没有出现在其他一些已知的出版物中，或者如果它本身对读者有价值，那么已经做出了相应的努力来呈现它。
