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In the first chapter, Kolmogorov’s axioms of the theory of probability are presented. The choice of the system of the axioms is explained in context of the Caratheodory theorem. Different probability spaces as well as probability distributions are introduced too. The famous Kolmogorov consistency theorem is formulated, and as a consequence, a brief construction of the Wiener measure is shown (see Baldi, An introduction through theory and exercises, 2017; Bulinski and Shiryayev, Theory of stochastic processes, 2005; Çinlar, Probability and stochastics, 2011; Durrett, Essentials of stochastic processes, 2018; Jacod and Protter, Probability essentials, 2003; Kolmogorov, Foundations of the theory of probability, 1956; Shiryaev, Probability, 1996, and Williams, Probability and martingales, 1991).

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Probabilistic Foundations

  • 章节

  • 发表日期：

    2026年

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  • Alexander Melnikov

摘要

In the first chapter, Kolmogorov’s axioms of the theory of probability are presented. The choice of the system of the axioms is explained in context of the Caratheodory theorem. Different probability spaces as well as probability distributions are introduced too. The famous Kolmogorov consistency theorem is formulated, and as a consequence, a brief construction of the Wiener measure is shown (see Baldi, An introduction through theory and exercises, 2017; Bulinski and Shiryayev, Theory of stochastic processes, 2005; Çinlar, Probability and stochastics, 2011; Durrett, Essentials of stochastic processes, 2018; Jacod and Protter, Probability essentials, 2003; Kolmogorov, Foundations of the theory of probability, 1956; Shiryaev, Probability, 1996, and Williams, Probability and martingales, 1991).