Absolute Continuity of Function on Topological Space using Measure

TL;DR

This paper generalizes the concept of absolute continuity to functions on topological measure spaces, developing new axioms, structures, and analytical properties that extend classical results beyond real spaces.

Contribution

It introduces the notion of topological measure spaces (tms), constructs new measures on second countable metric spaces, and explores absolute continuity in this broader setting.

Findings

Absolute continuity functions form a vector space and algebra over the field .

Established a relation between absolute continuity and locally Lipschitz functions on tms.

Characterized absolute continuity of linear maps via boundedness on separable normed linear spaces.

Abstract

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 $\sigma$- 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 $\sigma$-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{Carath\'eodory}'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.