Lecture2

Lecture2 - Lecture 2 : Ideas from measure theory 2.1...

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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 Dynkins form of the monotone class theorem is presented. Definition 2.1.1 Let be a set of points . In probability theory, represents all possible outcomes of an experiment or observation. Example 2.1.2 Tossing a coin has a set of outcomes = { Head,Tail } . Example 2.1.3 The position of a body in a 3-D Euclidean space belongs to the set = R 3 . A subset of 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 . Since well want to talk about combinations of events, a systematic treatment will require this class of subsets to have some nice set-theoretic properties. The next definition spells this out precisely. Definition 2.1.4 A class F of subsets of a space is called a field if it contains itself and is closed under complements and finite unions. That is 1. F 2. A F implies A c F...
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Lecture2 - Lecture 2 : Ideas from measure theory 2.1...

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