By Leadbetter R., Cambanis S., Pipiras V.

ISBN-10: 1107020409

ISBN-13: 9781107020405

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**Additional info for A Basic Course in Measure and Probability: Theory for Applications**

**Sample text**

Then μ(lim En ) = μ(∪∞ 1 Ei ) = μ{∪∞ 1 (Ei – Ei–1 )} = = = ∞ 1 μ(Ei – Ei–1 ) (the sets lim n1 μ(Ei – Ei–1 ) n→∞ lim μ{∪n1 (Ei – Ei–1 )} n→∞ (Ei – Ei–1 ) being disjoint and in R) = lim μ(En ), n→∞ as required. 5 If μ is a measure on a ring R, and {En } is a monotone decreasing sequence of sets in R, of which at least one has ﬁnite measure, and if lim En ∈ R, then μ(lim En ) = lim μ(En ). n→∞ Proof If μ(Em ) < ∞ then μ(En ) < ∞ for n ≥ m and μ(lim En ) < ∞ since lim En ⊂ Em . Now (Em – En ) is monotone increasing in n, and lim (Em – En ) = ∪n (Em – En ) = Em – ∩n En = Em – lim En ∈ R.

8 σ-rings, σ-ﬁelds and related classes 15 different closure operations and obtain a theorem of Sierpinski (popularized by Dynkin) to be used for such purposes (since this will require fewer restrictions on E than the assumption that it is a ring). Speciﬁcally we shall consider a nonempty class D which is closed under formation of both proper differences and countable disjoint unions4 (that is if E, F ∈ D and E ⊃ F, then E – F ∈ D and if Ei ∈ D, i = 1, 2, . . for disjoint Ei , then ∪∞ 1 Ei ∈ D).

Choose 0 < b0 > a0 ). Then (a0 , b0 ] ⊂ ∪∞ i=1 (ai , bi ] so that clearly < b0 – a0 (assuming i [a0 + , b0 ] ⊂ ∪∞ i=1 (ai , bi + /2 ). e. compactness), the bounded closed interval on the left is contained in a ﬁnite number of the open intervals on the right, and hence for some n, [a0 + , b0 ] ⊂ ∪ni=1 (ai , bi + /2i ). 2, b0 – a0 – ≤ n i=1 (bi Since is arbitrary, b0 – a0 ≤ – ai + ∞ i=1 (bi 2i ) ≤ ∞ i=1 (bi – ai ) + . – ai ), as required. 4 There is a unique measure μ on the σ-ﬁeld B of Borel sets, such that μ{(a, b]} = b – a for all real a < b.

### A Basic Course in Measure and Probability: Theory for Applications by Leadbetter R., Cambanis S., Pipiras V.

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