A Course in Probability Theory, Third Edition by Kai Lai Chung

By Kai Lai Chung

Because the e-book of the 1st variation of this vintage textbook over thirty years in the past, tens of millions of scholars have used A path in likelihood Theory. New during this version is an advent to degree thought that expands the marketplace, as this remedy is extra in keeping with present classes.

While there are numerous books on likelihood, Chung's booklet is taken into account a vintage, unique paintings in chance idea as a result of its elite point of sophistication.

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HINT: Prove first that there exists E with "arbitrarily small" probability. A quick proof then follows from Zorn's lemma by considering a maximal collection of disjoint sets, the sum of whose probabilities does not exceed a. ] *24. A point x is said to be in the support of a measure μ on &n iff every open neighborhood of x has strictly positive measure. The set of all such points is called the support of μ. Prove that the support is a closed set whose complement is the maximal open set on which μ vanishes.

Indeed the definition of the outer measure given above may be replaced by the equivalent one: (8) μ*{Ε) = Μ2μ(ϋη). n where the infimum is taken over all countable unions [J Un such that each n Un E &0 and [J Un => E. For another case where such a construction is n required see Sec. 3 below. m. v that corresponds to the given Fin the same way? It is important to realize that this question is not answered by the two preceding theorems. m. v that is defined on a domain strictly containing ^ 1 and that coincides with μ on ^ 1 (such as the μ on & as mentioned above) will certainly correspond to F in the same way, and strictly speaking such a v is to be considered as distinct from μ.

PROOF. We give the proof only in the case where μ and v are both finite, leaving the rest as an exercise. Let ν = {Εε^:μ(Ε) = ν(Ε)}, then Ή => J*o by hypothesis. But <€ is also a monotone class, for \îEnec€ for every « and En\ E ox En\ E, then by the monotone property of μ and v, respectively, /*(£) = lim μ(Εη) = lim Κ ^ ) = <Ε). 2 that *€ => J^, which proves the theorem. Remark. In order that /x and v coincide on «^ζ, it is sufficient that they coincide on a collection ^ such that finite disjoint unions of members of ^ constitute J^.

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