关于Banach和Kuratowski定理及广义强序列

中文摘要 AI 辅助

1929年，Banach和Kuratowski在CH（连续统假设）下证明了一个组合定理，该定理蕴含在实数集上不存在对每个实数集合都有定义的非零σ-加性有限测度。2003年，Bartoszyński和Halbeisen证明了Banach和Kuratowski定理等价于存在基数为连续统势的K-Lusin集合，且此类集合的存在性与$ZFC + \ eg CH$独立。另一方面，1965年Efimov引入了强序列方法，该方法用于证明二进空间中的一些著名定理。本文旨在表明所有这些概念都可以被推广，即在广义Baire空间中考虑，并展示它们之间的一些等价关系。此外，还展示了在calibres、有界性和划分关系方向上的若干应用。

英文摘要

In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing $σ$-additive finite measure on $\mathbb{R}$ which is defined for every set of reals. In 2003 Bartoszyński and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of $ZFC + \neg CH$. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.

发表机构

• Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）

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