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This chapter introduces and proves the Caratheodory Theorem and the Measure Extension Theorem. It is then shown in full generality that every non-decreasing, left-continuous function, such that \(F(-\infty ) = 0\) , \(F(\infty ) = 1\) is the cumulative distribution function of some probability measure. Finally, the Radon-Nikodym Theorem is presented without proof, followed by the definition and basic properties of the conditional expectation of random variables.

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Extension of Measure

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  • 发表日期：

    2025年

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  • Jolanta Misiewicz

摘要

This chapter introduces and proves the Caratheodory Theorem and the Measure Extension Theorem. It is then shown in full generality that every non-decreasing, left-continuous function, such that \(F(-\infty ) = 0\) , \(F(\infty ) = 1\) is the cumulative distribution function of some probability measure. Finally, the Radon-Nikodym Theorem is presented without proof, followed by the definition and basic properties of the conditional expectation of random variables.