Kolmogorov's Early Work on Convergence Theory and Foundation
Kolmogorov's Early Work on Convergence Theory and Foundation
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科尔莫哥洛夫关于收敛理论和基础的早期工作
DOI：
10.1214/aop/1176991247
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发表时间：
1989
影响因子：
2.3
通讯作者：
J. Doob
中科院分区：
文献类型：
--
作者：
J. Doob
1.在Kolmogorov之前的概率。当哥洛夫开始他的数学生涯，非数学概率，因为它仍然是，研究各种不是非常精确定义的真实的背景。其中一些背景引起了数学问题，例如在组合学中，但不清楚总体数学概率背景是什么，或者是否可能。庞加莱曾在1912年写道[13]“在ne peut donner une definition satisfaisante de la probabilite.这是典型的写作的时间，事实上相当近的时间，读者不能肯定是否作家是思考概率作为一个非数学或数学主题在他的声明。冯·米塞斯在1919年[15]更清楚地表达了他的痛惜，但也同样悲观，尽管更专业化地进行了沉思：“在达特看来，现在的战争不应该是一个数学问题。冯·米塞斯试图创建他所期望的数学学科，但他的“集体”理论是一个混乱的，但暗示混合数学和非数学的背景。鉴于哥洛夫的高度评价冯米塞斯一些解释性的意见是适当的。考虑一个序列，它是通过对具有共同分布的一系列独立试验进行抽样而得到的。(Note对一个无限序列进行采样是这种分析的一个不切实际的元素。）存在与这样的序列相关联的典型性质，例如由大数定律所指示的。冯·米塞斯试图通过这些典型性质的形式化，通过构建具有足够这些性质的个体序列作为试验序列的模型，来构建数学和非数学概率的基础。他最初对这样一个个体序列的定义（一个“集体”）[15]，对采样中似乎发生的事情有深刻的见解，但当应用于个体序列时，要么是空洞的，要么是毫无意义的，这取决于读者对冯·米塞斯的话的解释。他后来的定义[16]有太少的属性是有用的。在任何情况下，这样的构造，即使成功了，显然也太笨拙和太有限，不能被认真地考虑作为现代概率数学分析的非凡范围的有用基础。另一方面，这种序列的形式化是Kolmogorov在几个场合讨论过的一个吸引人的概念问题，
1. Probability before Kolmogorov. When Kolmogorov was starting his mathematical career, nonmathematical probability was, as it still is, the study of various not very precisely defined real contexts. Some of these contexts gave rise to mathematical problems, in combinatorics for example, but it was not clear what an overall mathematical probability context would be or indeed whether one was possible. Poincare had written in 1912 [13] "On ne peut donner une definition satisfaisante de la probabilite." It was typical of the writing of that time, and in fact of considerably more recent times, that the reader could not be certain whether the writer was thinking of probability as a nonmathematical or a mathematical subject in his statement. von Mises in 1919 [15] was clearer in what he deplored but was just as pessimistic, although more professorily ponderous: "In der Tat kann man den gegenwirtigen Zustand kaum anders als dahin kennzeichnen, dass die Wahrscheinlichkeitsrechnung heute ein mathematische Disciplin nicht ist." von Mises attempted to create his desired mathematical discipline but his theory of "collectives" was a confused although suggestive mixture of mathematical and nonmathematical contexts. In view of Kolmogorov's high opinion of von Mises a few explanatory remarks are appropriate here. Consider a sequence obtained by sampling a sequence of independent trials with a common distribution. (Note that sampling an infinite sequence is an unrealistic element of this analysis.) There are typical properties associated with such a sequence, as indicated for example by the law of large numbers. von Mises attempted to construct a basis for mathematical as well as nonmathematical probability by a formalization of these typical properties, by constructing an individual sequence with enough of these properties to be a model for a trial sequence. His original definition of such an individual sequence (a "collective") [15], was insightful of what seems to happen in sampling, but was either vacuous or meaningless when applied to an individual sequence, depending on the reader'sinterpretation of von Mises' words. His later definition [16] had too few properties to be useful. In any event such a construction, even if successful, would obviously be too awkward and too limited to be considered seriously as a useful basis for the extraordinary scope of modem probabilistic mathematical analysis. On the other hand, the formalization of such a sequence is an appealing conceptual problem which Kolmogorov discussed on several occasions, most