Cardinal algebras and measures invariant under equivalence relations.
Cardinal algebras and measures invariant under equivalence relations.
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等价关系下的基数代数和测度不变。
DOI：
10.1090/s0002-9947-1969-0245743-5
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发表时间：
1969
影响因子：
1.3
通讯作者：
R. Chuaqui
中科院分区：
文献类型：
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作者：
R. Chuaqui
导论.有讨论从时间到时间的“抽象措施”的价值观，其中不需要是数字（例如[2]，[3]，[4]，[6]，[7]）。本文的目的之一是提出赞成使用基数代数作为这些措施的价值观的论点。基数代数是由A. Tarski在[8]中。它们具有许多真实的数的良好性质，并且在以下情况下自然出现：一个（伪）群G的一一函数是给定的域和范围在一个u-环的集合X2之间的等价关系1的成员定义如下：A B如果有Ai，Bi E 1[，fi E G对于i<co使得Ai n Aj=O=B=i r Bj对于i#Oj，A=JUI <。Ai，B=Ui<co Bi，Ai（Domfi）andfi*（Ai）=Bi对所有i<co，这是可数分解的等价性. [2]、[3]、[4]、[6]、[7]中考虑了类似的等价关系。当目标是获得一个忠实于这种形式的等价关系的测度时，第一个自然的步骤是考虑由关系确定的等价类。碰巧这些等价类与适当定义的有限和无限加法形成一个广义基数代数。例如，[4]和[6]中考虑的“测度代数”是广义基数代数。我的第二个目的是确定在什么条件下我们可以得到一个忠实于等价关系的数值测度，也就是说，一个满足以下条件的可数可加测度：（a）测度为零的集合是那些必须有测度为零的集合，也就是说，在测度为1的集合中包含无穷多个不相交等价集合的集合。我称这些集合为可忽略的。(b)对于具有正测度的集合，两个集合具有相同测度当且仅当它们等价。这些特征对于概率测度特别重要，在概率测度中，我们希望尽可能忠实于等似然关系（cf. [3]）。下面的定理2.11给出了获得这样一个测度的等价关系的充分（几乎是必要的）条件。
Introduction. There have been discussions from time to time of "abstract measures" the values of which need not be numerical (e.g. [2], [3], [4], [6], [7]). One of the purposes of this paper is to present arguments in favor of the use of cardinal algebras as values for these measures. Cardinal algebras were introduced and developed by A. Tarski in [8]. They have many of the good properties of real numbers and arise naturally in situations like the following: A (pseudo) group G of one-one functions is given with domain and range in a u-ring of sets X2 An equivalence relation between members of 1 is defined as follows: A B iffthere are Ai, Bi E 1[,fi E G for i<co such that Ai n Aj=O=B=i r Bj for i#Oj, A=JUi < . Ai, B=Ui<oo Bi, Ai(Domfi andfi*(Ai)=Bi for all i<co. This is equivalence by countable decomposition. Equivalence relations like have been considered in [2], [3], [4], [6], [7]. When the aim is to obtain a measure that is faithful to an equivalence relation of this form, the first natural step is to consider the equivalence classes determined by the relation. It happens that these equivalence classes with suitably defined finite and infinite addition form a generalized cardinal algebra. For instance, the "measure algebras" considered in [4] and [6] are generalized cardinal algebras. My second purpose is to determine in what conditions we can obtain a numerical measure faithful to the equivalence relation; that is, a countably additive measure that satisfies the following: (a) The only sets with measure zero are those that necessarily have to have it. That is, sets that have infinitely many disjoint equivalent sets contained in a set of measure one. These sets I call negligible. (b) For sets with positive measure, it should be valid that two sets have the same measure iff they are equivalent. These characteristics are specially important with respect to probability measures where we want to be as faithful as possible to the equal likelihood relation (cf. [3]). Theorem 2.11, below, gives sufficient (and almost necessary) conditions on the equivalence relation to obtain such a measure.