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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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