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Extra info for Seminaire de Probabilites XVIII 1982 83
Expectation. 1 General deﬁnitions Let the probability space , F , P be given. R1 D 1, C1 the (ﬁnite) Ł real line, R D [ 1, C1] the extended real line, B1 D the Euclidean Borel ﬁeld on R1 , BŁ D the extended Borel ﬁeld. A set in BŁ is just a set in B possibly enlarged by one or both points š1. DEFINITION OF A RANDOM VARIABLE. A real, extended-valued random variable is a function X whose domain is a set 1 in F and whose range is contained in RŁ D [ 1, C1] such that for each B in BŁ , we have 1 fω: X ω 2 Bg 2 1 \ F where 1 \ F is the trace of F on 1.
Since lim sup Xj D inf sup Xj , j n j½n 40 RANDOM VARIABLE. EXPECTATION. 1 Xj exists [and is ﬁnite] on the set where lim supj Xj D lim infj Xj [and is ﬁnite], which belongs to F , the rest follows. v. given at the beginning of this section. DEFINITION. v. X is called discrete (or countably valued ) iff there is a countable set B ² R1 such that P X 2 B D 1. f. is. v. need not have a range that is discrete in the sense of Euclidean topology, even apart from a set of probability zero. v. f. in Example 2 of Sec.
4) we note the following extension. Let P be deﬁned on a ﬁeld F which is ﬁnitely additive and satisﬁes axioms (i), (iii), and (1). Then (ii) holds whenever k Ek 2 F . For then 1 kDnC1 Ek also belongs to F , and the second part of the proof above remains valid. The triple , F , P is called a probability space (triple); alone is called the sample space, and ω is then a sample point. F. F on 1 is the collection of all sets of the form 1 \ F, where F 2 F . F. of subsets of 1, and we shall denote it by 1 \ F .