Nikodym性质与$ω$上的滤子
The Nikodym property and filters on $ω$
AI总结：
本文研究ω上滤子对应的拓扑空间与布尔代数Nikodym性质的关联，刻画了AN类理想的等价条件，分析了其Katětov序结构及密度理想的相关等价性质。
摘要：
对于$ω$上的自由滤子$F$，令$N_F=ω\cup\{p_F\}$（其中$p_F\notinω$），并赋予如下拓扑：$ω$中的每个元素都是孤立点，而$p_F$的所有开邻域都形如$A\cup\{p_F\}$，其中$A\in F$。本文旨在在布尔代数的Nikodym性质背景下研究形如$N_F$的空间。我们用$\mathcal{AN}$表示$ω$上所有满足如下条件的理想$\mathcal{I}$构成的类：对于其对偶滤子$\mathcal{I}^*$，空间$N_{\mathcal{I}^*}$上存在一列有限支撑的带号测度$\langleμ_n\colon n\inω\rangle$，使得$\\|μ_n\\|\rightarrow\infty$，且对每个既开又闭的子集$A\subseteq N_{\mathcal{I}^*}$都有$μ_n(A)\rightarrow 0$。我们证明，$\mathcal{I}\in\mathcal{AN}$当且仅当存在$ω$上的密度子测度$φ$满足$φ(ω)=\infty$且$\mathcal{I}$包含于穷竭理想$\mbox{Exh}(φ)$。由此可得：若存在$ω$上的密度子测度$φ$满足$φ(ω)=\infty$使得$\mathcal{I}\subseteq\mbox{Exh}(φ)$，且$N_{\mathcal{I}^*}$同胚于给定布尔代数$\mathcal{A}$的Stone空间$St(\mathcal{A})$的一个子空间，则$\mathcal{A}$不具有Nikodym性质。我们观察到每个$\mathcal{I}\in\mathcal{AN}$都在Katětov序下低于渐近密度零理想$\mathcal{Z}$，并证明$\mathcal{AN}$类存在一个大小为$\mathfrak{d}$的子集，该子集关于Katětov序$\leq_K$是支配的，但$\mathcal{AN}$不存在$\leq_K$-极大元。我们还证明，对于密度理想$\mathcal{I}$，$\mathcal{I}\notin\mathcal{AN}$当且仅当$\mathcal{I}$是完全有界的，当且仅当布尔代数$\mathcal{P}(ω)/\mathcal{I}$包含一个可数分裂族。
英文摘要：
For a free filter $F$ on $ω$, let $N_F=ω\cup\{p_F\}$, where $p_F\not\inω$, be equipped with the following topology: every element of $ω$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. The aim of this paper is to study spaces of the form $N_F$ in the context of the Nikodym property of Boolean algebras. By $\mathcal{AN}$ we denote the class of all those ideals $\mathcal{I}$ on $ω$ such that for the dual filter $\mathcal{I}^*$ the space $N_{\mathcal{I}^*}$ carries a sequence $\langleμ_n\colon n\inω\rangle$ of finitely supported signed measures such that $\|μ_n\|\rightarrow\infty$ and $μ_n(A)\rightarrow 0$ for every clopen subset $A\subseteq N_{\mathcal{I}^*}$. We prove that $\mathcal{I}\in\mathcal{AN}$ if and only if there exists a density submeasure $φ$ on $ω$ such that $φ(ω)=\infty$ and $\mathcal{I}$ is contained in the exhaustive ideal $\mbox{Exh}(φ)$. Consequently, we get that if $\mathcal{I}\subseteq\mbox{Exh}(φ)$ for some density submeasure $φ$ on $ω$ such that $φ(ω)=\infty$ and $N_{\mathcal{I}^*}$ is homeomorphic to a subspace of the Stone space $St(\mathcal{A})$ of a given Boolean algebra $\mathcal{A}$, then $\mathcal{A}$ does not have the Nikodym property. We observe that each $\mathcal{I}\in\mathcal{AN}$ is Katětov below the asymptotic density zero ideal $\mathcal{Z}$, and prove that the class $\mathcal{AN}$ has a subset of size $\mathfrak{d}$ which is dominating with respect to the Katětov order $\leq_K$, but $\mathcal{AN}$ has no $\leq_K$-maximal element. We show that for a density ideal $\mathcal{I}$ it holds $\mathcal{I}\not\in\mathcal{AN}$ if and only if $\mathcal{I}$ is totally bounded if and only if the Boolean algebra $\mathcal{P}(ω)/\mathcal{I}$ contains a countable splitting family.
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