论文标题
lebesgue测量自然数的零模型理想
Lebesgue measure zero modulo ideals on the natural numbers
论文作者
论文摘要
我们建议对Lebesgue的理想$ \ MATHCAL {N} $进行重新制定。以同样的方式，我们重新制定了由$f_σ$生成的理想$ \ mathcal {e} $，零集的零集Modulo $ j $，我们用$ \ Mathcal {n}^*_ J $表示。我们表明，这些是$σ$ -IDEALS，并且$ \ MATHCAL {N} _J = \ MATHCAL {N} $ IFF $ J $具有BAIRE属性，这又等同于$ \ Mathcal {n}^*_ J = _ J = _ j = \ Mathcal {e} $。此外，我们证明$ \ MATHCAL {N} _J $不包含共享收益集，而$ \ Mathcal {n}^*_ j $当$ j $没有baire属性时包含非锻炼集。我们还证明了这些理想模式$ j $与几乎连贯的过滤器（或理想）的概念之间的联系。 我们还研究了与$ \ Mathcal {n} _J $和$ \ Mathcal {n}^*_ J $相关的基本特征。我们展示了他们在Cichoń的图表方面的立场，并证明了与连续体的其他非常经典的主要基本特征相关的一致性结果，仅留下了很少的开放问题。为此，我们发现了$ \ mathrm {add}（\ mathcal {n}）$和$ \ mathrm {cof}（\ mathcal {n}）$的新特征。我们还表明，在Cohen模型中，我们可以获得与我们的新理想相关的基本特征的许多不同值。
We propose a reformulation of the ideal $\mathcal{N}$ of Lebesgue measure zero sets of reals modulo an ideal $J$ on $ω$, which we denote by $\mathcal{N}_J$. In the same way, we reformulate the ideal $\mathcal{E}$ generated by $F_σ$ measure zero sets of reals modulo $J$, which we denote by $\mathcal{N}^*_J$. We show that these are $σ$-ideals and that $\mathcal{N}_J=\mathcal{N}$ iff $J$ has the Baire property, which in turn is equivalent to $\mathcal{N}^*_J=\mathcal{E}$. Moreover, we prove that $\mathcal{N}_J$ does not contain co-meager sets and $\mathcal{N}^*_J$ contains non-meager sets when $J$ does not have the Baire property. We also prove a deep connection between these ideals modulo $J$ and the notion of nearly coherence of filters (or ideals). We also study the cardinal characteristics associated with $\mathcal{N}_J$ and $\mathcal{N}^*_J$. We show their position with respect to Cichoń's diagram and prove consistency results in connection with other very classical cardinal characteristics of the continuum, leaving just very few open questions. To achieve this, we discovered a new characterization of $\mathrm{add}(\mathcal{N})$ and $\mathrm{cof}(\mathcal{N})$. We also show that, in Cohen model, we can obtain many different values to the cardinal characteristics associated with our new ideals.
