# Real Analysis and Probability by Robert B. Ash

By Robert B. Ash

This e-book, the 1st of a projected quantity sequence, is designed for a graduate path in glossy likelihood. the 1st 4 chapters, besides the Appendix: On normal Topology, give you the historical past in research wanted for the examine of likelihood. This fabric is accessible as a separate ebook referred to as" degree, Integration, and practical Analysis."

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Additional resources for Real Analysis and Probability

Example text

Let ffi",, be the class of sets A u N, where A ranges over ffi" and N over all subsets of sets of measure 0 in ffi". Now ffi",, is a a-field including~. for it is clearly closed under countable union, and if A u NE ffi", N c ME ffi", µ(M) = 0, then (A u NY = Ac n Nc = (Ac n Mc) u (Ac n (Nc - Mc)) and Ac n (Nc - Mc) =Ac n (M - N) c M, so (A u NYE ffi",,. We extendµ to ffi",, by setting µ(A u N) =µ(A). This is a valid definition, for if A 1 u N 1 =Aiu Ni effi",,, we have since A 1 - Ai c Ni. Thus µ(A 1) ~µ(Ai), and by symmetry, µ(A 1) = µ(A 2 ).

We have therefore shown that :If is a field. Now equality holds in (I), for if not, the sum of the left sides of (I) and (2) would be less than the sum of the right sides, a contradiction. Thusµ* is finitely additive on :If. To show that :If is a er-field, let Hn E :If, n = 1, 2, ... 3(b). 3(d), hence for any e > 0, µ*(H) ::;; µ*(Hn) + e for large n. 3(c), and Hn E :If, we have µ*(H) + µ*(W)::;; I + e. Since e is arbitrary, He :If, making :If a er-field. 8(a). 6 Theorem. A finite measure on a field on a(~ 0 ).

3 EXTENSION OF MEASURES 19 PROOF. Jff = F,.. where F = u(F0 ). Jff, by definition of µ*(A) and µ*(A') we can find sets Gn, Gn' e u(fF 0 ), n = I, 2, ... µ*(A). Let G = U:°=1 Gn, G' = Gn'· ThenA =Gu (A - G),G e u(fF 0 ),A - G c G' - Ge u(F 0 ), µ*(G' - G) :5; µ*(Gn' - Gn)-.. 0, so that µ*(G' - G) = 0. F,. •. , then B =A u N, A e F, N c Me IF, µ*(M) = 0. Jff. Jff. 9 Monotone Class Theorem. Let F 0 be a field of subsets of n, and re a class of subsets of n that is monotone (if An E re and An i A or An !