Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions

Abstract

In this article our main object of investigation is the simple modular density ideals ${\mathcal{Z}}_g(f)$ introduced in [Bose et al., Indag. math., 2018] where $g$ is a weight function, more precisely, $g\in G$, $G=\{g:\ddot{I}‰\to [0,\infty ):\frac{k}{g(k)}\nrightarrow 0\text{ and }g(k)\to \infty \text{ as }k\to \infty \}$ and $f$ is an unbounded modulus function. We mainly investigate certain properties of these ideals in line of [Kwela et al, J. math. Anal. Appl., 2019]. For an unbounded modulus function $f$ it is shown that there are $1$ or $\ck$ many functions $g\in G$ generating the same ideal ${\mathcal{Z}}_g(f)$. We then obtain certain interactive results involving the sequence of submeasures $\{\ddot{I}{•}_k{\}}_{k\in \ddot{I}‰}$ generating the ideal ${\mathcal{Z}}_g(f)$ and the functions $g,f$. Finally, we present some observations on ${\mathcal{Z}}_g(f)$ ideals related to the notion of increasing-invariance.