密度零理想的推广
A generalization of the density zero ideal
摘要：
设$\mathscr{F}=(F_n)$是$ω$的非空有限子集序列，使得$\lim_n |F_n|=\infty$和定义理想$F_n=\{1,\ldots,n\}$。证明了$\mathcal{I}(\mathscr{F})$是解析P-理想，但不是$F_σ$ .因此，我们证明了实有界序列的集合是$\mathcal{I}(\mathscr{F})$ -收敛于$0$的，在$\ell_\infty$中是不可补的.
英文摘要：
Let $\mathscr{F}=(F_n)$ be a sequence of nonempty finite subsets of $ω$ 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 $F_n=\{1,\ldots,n\}$ corresponds to the classical case of density zero ideal. We show that $\mathcal{I}(\mathscr{F})$ is an analytic P-ideal but not $F_σ$. As a consequence, we show that the set of real bounded sequences which are $\mathcal{I}(\mathscr{F})$-convergent to $0$ is not complemented in $\ell_\infty$.
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