泛型扩张中的Universally measurable集
Universally measurable sets in generic extensions
摘要：
一个拓扑空间的子集称为\emph{universally measurable}，如果它是由空间上的每个可数可加$σ$ -有限Borel测度的完备化来度量的，而\emph{universally null}如果它对每个这样的无原子测度都有测度零。在1908年，Hausdorff证明了存在基数为$\aleph_{1}$的实数的泛空集，因此至少存在$2^{\aleph_{1}}$这样的集合。拉弗在20世纪70年代证明了一致地存在连续统的许多普遍空的实数集。这个问题是否存在超过连续许多普遍可测的实数集是问Mauldin在1978年。我们表明，一贯存在只有连续许多普遍可测的集。这个结果也来自Ciesielski和Pawlikowski对迭代Sacks模型的研究。在模型中，我们考虑（适当大小的随机代数的力迫扩张）每一个实数集是普遍可测的当且仅当它和它的补是许多Borel集的基础模型连续统的并集。
英文摘要：
A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $σ$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets.
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