论文标题
使用测量的拓扑空间上功能的绝对连续性
Absolute Continuity of Function on Topological Space using Measure
论文作者
论文摘要
本文的主要目的是在$ \ r $之外的更通用的设置上开发功能绝对连续性的概念。为此，我们考虑了一个拓扑空间，这也是一个测量空间。我们已经建立了用于制造$σ$ - 代数并与空间拓扑兼容的公理。这些空间被称为\ textit {拓扑度量空间}（在简称\ textit {tms}中）。 $ \ r^n $带有通常的拓扑结构，lebesgue $σ$ -Algebra和Lebesgue Measure是TMS的相关示例。此外，我们在第二个可数度度空间上提出了新的TMS结构，并开发了一种新措施。这种结构是由\ textbf {carathéodory}的定理激励的。在这个新的TMS框架中，我们不仅在$ \ r^n（n \ geq 2）$上，还可以在任何可分开的规范线性空间上探索绝对连续功能的充分收集。此外，我们已经描述了一些分析方面，这些方面具有TMS框架上绝对连续性的内在含义。此外，在TMS上的所有绝对连续功能的集合形成了$ \ k $，真实或复数字段以及额外有限属性的矢量空间，它们在$ \ k $上形成ring and Algebra。此后，我们在TMS上介绍了\ textIt {locally Lipschitcz函数}的概念，该函数涉及开放连接集的测量。已经建立了绝对连续性与局部Lipschitz之间的关系。我们已经证明，线性函数的绝对连续性和界限在可分离的规范线性空间与该新度量的关联上是相同的。此外，我们已经将绝对连续函数的共同域扩展到规范的线性空间，这有助于我们以界限的绝对连续性来表征界限的绝对连续性，当时该域是与该新度量融合在一起的可分开的规范线性空间。
The prime objective of this paper is to develop the notion of absolute continuity of functions on a more general setting outside $\R$. For this we have considered a topological space which is a measure space as well. We have built axioms for making the $σ$- algebra and measure compatible with the topology of the space. These spaces are termed as \textit{topological measure space} (in short \textit{tms}). $\R^n$ with usual topology, Lebesgue $σ$-algebra and Lebesgue measure is a relevant example of tms. Further, we have presented a new tms structure on second countable metric spaces with the development of a new measure. This construction is motivated by \textbf{Carathéodory}'s Theorem. In this new tms framework, we have accomplished exploring ample collection of absolutely continuous functions not only on $\R^n(n\geq 2)$ but also on any seperable normed linear space. Also, we have described several analytical aspects carrying the intrinsic sense of absolute continuity on tms framing. Besides, the collection of all absolutely continuous functions on tms forms a vector space over $\K$, the field of real or complex numbers and with additional boundedness property, they form ring and algebra over $\K$. Thereafter, we have introduced the concept of \textit{locally Lipschitcz function} on tms involving the measurement of open connected sets. A relation between absolute continuity and locally Lipschitz has been developed. We have proved that absolute continuity and boundedness of linear functionals are same on separable normed linear spaces with the association of that new measure. Further, we have extended the co-domain of absolutely continuous functions upto normed linear spaces which helps us to characterise absolute continuity of linear maps in terms of boundedness when the domain is a seperable normed linear space incorporated with that new measure.
