On some topology generated by $\mathcal{I}$-density function

TL;DR

This paper investigates the properties of the $ ext{I}$-density function and introduces a new topology on the real numbers based on this function, providing a characterization of Lebesgue measurable sets within this framework.

Contribution

It develops the $ ext{I}$-density topology on reals and characterizes Lebesgue measurable sets using Borel sets in this topology.

Findings

Properties of $ ext{I}$-density function are established.

A new topology on reals is induced by $ ext{I}$-density.

Lebesgue measurable sets are characterized via Borel sets in the $ ext{I}$-density topology.

Abstract

In this paper we have studied on $\mathcal{I}$-density function using the notion of $\mathcal{I}$-density, introduced by Banerjee and Debnath \cite{banerjee 4} where $\mathcal{I}$ is an ideal of subsets of the set of natural numbers. We have explored certain properties of $\mathcal{I}$-density function and induced a topology using this function in the space of reals namely $\mathcal{I}$-density topology and we have given a characterization of the Lebesgue measurable subsets of reals in terms of Borel sets in $\mathcal{I}$-density topology.