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Demystifying measure-theoretic probability theory (part 3: expectation) - Matthew N. Bernstein

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In this series of posts, I present my understanding of some basic concepts in measure theory — the mathematical study of objects with “size”— that have enabled me to gain a deeper understanding into the foundations of probability theory. In part 3, we will build from the measure-theoretic definitions for probability and random variables towards the measure-theoretic definition for the expectation of a random variable. So far, we have laid out some foundational definitions for a measure-theoretic treatment of probability. By doing so, we have unified the concepts of discrete random variables and continuous random variables, as are often taught in introductory courses. Furthermore, this rigorous definition of random variables can describe non-numeric random variables. Now, we will discuss how expectation is defined for the more rigorous, measure-theoretic definition of a random variable. First, let’s review the basic notion for the expected value of a random variable. Intuitively, the ex

In this series of posts, I present my understanding of some basic concepts in measure theory — the mathematical study of objects with “size”— that have enabled me to gain a deeper understanding into the foundations of probability theory. In part 3, we will build from the measure-theoretic definitions for probability and random variables towards the measure-theoretic definition for the expectation of a random variable. Introduction So far, we have laid out some foundational definitions for a measure-theoretic treatment of probability. By doing so, we have unified the concepts of discrete random

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