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Unformatted text preview: Lecture 2 : Ideas from measure theory 2.1 Probability spaces This lecture introduces some ideas from measure theory which are the foundation of the modern theory of probability. The notion of a probability space is defined, and Dynkin’s form of the monotone class theorem is presented. Definition 2.1.1 Let Ω be a set of points ω . In probability theory, Ω represents all possible outcomes of an experiment or observation. Example 2.1.2 Tossing a coin has a set of outcomes Ω = { Head,Tail } . Example 2.1.3 The position of a body in a 3D Euclidean space belongs to the set Ω = R 3 . A subset of Ω is called an event. It is natural to ask questions such as whether or not an outcome of a random experiment belongs to to a event. To do this, we need to define the events under consideration – we need to define a class of subsets of the space Ω. Since we’ll want to talk about combinations of events, a systematic treatment will require this class of subsets to have some nice settheoretic properties. The next definition spells this out precisely. Definition 2.1.4 A class F of subsets of a space Ω is called a field if it contains Ω itself and is closed under complements and finite unions. That is 1. Ω ∈ F 2. A ∈ F implies A c ∈ F...
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
 Spring '09
 Reed
 Probability

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