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In this chapter, a special attention is devoted to the absolute continuityAbsolute continuity of measures. It is shown how this notion and the Radon-Nikodym theorem work to define conditional expectations. The list of properties of conditional expectations is given here. In particular, it is emphasized the optimality in the mean-square sense of conditional expectations (see Çinlar, Probability and stochastics, 2011; Jacod and Protter, Probability essentials, 2003; Kolmogorov, Foundations of the theory of probability, 1956; Shiryaev, Probability, 1996; Skorokhod, Lectures on the theory of stochastic processes, 1996, and Williams, Probability and martingales, 1991).

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Absolute Continuity of Probability Measures and Conditional Expectations

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  • 发表日期：

    2026年

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  • Alexander Melnikov

摘要

In this chapter, a special attention is devoted to the absolute continuityAbsolute continuity of measures. It is shown how this notion and the Radon-Nikodym theorem work to define conditional expectations. The list of properties of conditional expectations is given here. In particular, it is emphasized the optimality in the mean-square sense of conditional expectations (see Çinlar, Probability and stochastics, 2011; Jacod and Protter, Probability essentials, 2003; Kolmogorov, Foundations of the theory of probability, 1956; Shiryaev, Probability, 1996; Skorokhod, Lectures on the theory of stochastic processes, 1996, and Williams, Probability and martingales, 1991).