Exh subspaces of bounded variation spaces generated by submeasures

Abstract

In the present paper we are building one more bridge between the theory of ideals on the set of natural numbers and the theory of spaces of functions of bounded variation. It is already known that $F_{\sigma }$ ideals (that is, ideals of the form $\text{Fin}(\varphi )$, where $\varphi$ is a lower semicontinuous submeasure on $\mathbb{N}$) lead to a nice generalization of different concepts from the theory of functions of bounded variation (Waterman variation and Chanturya classes). In this paper we show that analytic P-ideals (that is, ideals of the form $\text{Exh}(\varphi )$, where $\varphi$ is a lower semicontinuous submeasure on $\mathbb{N}$) lead to a generalization of spaces of functions continuous in $\Lambda$-variation. The properties of this generalization are investigated.