A generalization of the density zero ideal

Abstract

Let $\mathcal{F}=(F_n)$ be a sequence of nonempty finite subsets of $\omega$ such that ${\lim }_n|F_n|=\infty$ and define the ideal $$\mathcal{I}(\mathscr{F}):=\left\{A\subseteq ω: |A\cap F_n|/|F_n|\to 0~\mbox{as}~n\to \infty \right\}.$$ The case $F_n=\{1,\dots ,n\}$ corresponds to the classical case of density zero ideal. We show that $\mathcal{I}(\mathcal{F})$ is an analytic P-ideal but not $F_{\sigma }$. As a consequence, we show that the set of real bounded sequences which are $\mathcal{I}(\mathcal{F})$-convergent to $0$ is not complemented in ${\ell }_{\infty }$.

3 pages