Probabilities and Set Theory
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We say that an event A E A is nearly certain if A is nearly certainly equal to OMEGA. In other words, OMEGA = AUN , where N is a negligeable set.
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A proof involving probability and set theory is provided.
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In probability, all the possible outcomes of an experiment are collected in a set called the sample space. If Omega is used to represent the sample space, we can consider certain subsets of Omega to define events. We create a set F of subsets of Omega and require certain properties of those subsets to hold, thus creating a "sigma-algebra" of subsets of Omega.
The sigma-algebra, F, by definition has the properties that:
1) The empty set is an element of F
2) If E is an element of X, then so is the complement of E.
3) If E1, E2, ... is a sequence in F, then their countable union is in F.
The pair (Omega, F) is called a measureable space, being a set and a sigma-algebra over that set.
A measure, mu, is a function defined from a sigma-algebra, like F, to the set of ...
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