Measure and category in effective descriptive set theory
Measure and category in effective descriptive set theory
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DOI：
10.1016/0003-4843(73)90012-0
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发表时间：
1973-07
期刊：
影响因子：
--
通讯作者：
A. Kechris
中科院分区：
文献类型：
--
作者：
A. Kechris
我们在本文中关注的是连续统上的测度和范畴理论的一些可定义性方面。我们的研究对象是从测度论和拓扑的角度来看实数的射影子集及其结构。我们在这里解决的典型问题和我们证明的一些结果如下：（A）计算实数分析集的度量的复杂性（就分析层次而言）。我们证明 Z l 组实数的测度是 Z~ 实数。我们在这里假设存在可测量的cardi~,~ al（如果n--2）和射影确定性（如果n> 2）。这也适用于下面所述的定理。（B）有效近似~这里的代表性结果是a~ nl集合（n偶数）或I1~集合（n奇数）可以通过a]它的子集“任意接近”来近似。
We are concerned in this paper with some definability aspects of the theory of measure and category on the continuum. Our objects of study are the projective subsets of the reals and their structure from a measure theoretic and topological point of view. Typical problems we attack here and some of the results we prove are the following:(A) Computation of the complexity (in terms of the analytical hierarchy) of the measure of an analytical set of reals. We prove that the measure of a Z l set of reals is a Z~ real. We assume here the existence of a measurable cardi~,~ al if n--2 and Projective Determinacy if n> 2. This applies also to the theorems stated below.(B) Effective approximations~ A representative result here is that a~ nl set (n even) or I1~ set (n odd) can be approximated'arbitrarily close'by a A] a subset of it.