In fact in these notes whenever the sample space is

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Unformatted text preview: ∪ B ∈ F ). More generally, if A1 , A2 , . . . is a list of events then the union of all of these events (the set of outcomes in at least one of them), A1 ∪ A2 ∪ · · · , is also an event. One choice of F that satisfies the above axioms is the set of all subsets of Ω. In fact, in these notes, whenever the sample space Ω is finite or countably infinite (which means the elements of Ω can be arranged in an infinite list, indexed by the positive integers), we let F be the set of all subsets of Ω. When Ω is uncountably infinite, it is sometimes mathematically impossible to define a suitable probability measure on the set of all subsets of Ω in a way consistent with the probability axioms below. To avoid such problems, we simply don’t allow all subsets of such an Ω to be events, but the set of events F can be taken to be a rich collection of subsets of Ω that includes any subset of Ω we are likely to encounter in applications. If the Axioms E.1-E.3 are satisfied, the set of ev...
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