Set-theoretic study of measures

区分不可数与可数的关键事实是，平稳性的概念可以定义为不可数集合的子集。粗略地说，平稳性是具有正度量的组合对应物。P_κλ的组合学在其最普遍的意义上研究了这个概念。这里κ表示正则不可数基数，λ是至少κ的基数。当κ为λ-超紧基数时，在P_κλ上的正常超滤是研究的重要对象。如果κ是λ-超紧的，那么P_κλ是否携带一个性质为χ的正规超滤子一直是一个悬而未决的问题。在b[3]中，我们构造了一个性质稍弱的。证明了广义连续统假设暗示了具有χ性质的超滤波器的存在性。后一个结果在相对一致性结果中应该是有用的。在与Shelah b[2]的合作中，我们证明了当κ大于ω1且λ大于κ时，P_κλ的平稳反射原理失效。此外，我们在应用平稳反射原理方面也取得了进展。在[5]中，我们对Diamond原理在P_<ω1>λ上成立的结论作了两个推广。在[5]中，我们推广了Foreman-Magidor-Shelah关于P_<ω1>λ上的club滤波器可以饱和的著名结果。

The crucial fact that separates the uncountable from the countable is that the notion of stationariness can be defined for subsets of uncountable sets. Roughly speaking, stationariness is a combinatorial counterpart of having a positive measure.The combinatorics of P_κλ investigates that notion in its greatest generality. Here κ denotes a regular uncountable cardinal and λ is a cardinal at least κ.When κ, is a λ-supercompact cardinal, a normal ultrafilter on P_κλ is an important object to study. It has been an open question whether P_κλ carries a normal ultrafilter with property χ if κ, is λ-supercompact. In [3] we constructed one with a property slightly weaker than this. The proof shows in fact that the Generalized Continuum Hypothesis implies the existence of an ultrafilter with property χ. The latter result should be useful in relative consistency results.In joint work with Shelah [2] we showed that the principle of Stationary Reflection in P_κλ fails if κ is greater than ω1 and λ is greater than κ. Moreover we have made progress in applying the principle of Stationary Reflection. In [4] we gave two extensions of Shelah's result that the Diamond principle holds in P_<ω1>λ In [5] we extended the well-known result of Foreman-Magidor-Shelah that the club filter on P_<ω1>λ can be presaturated.