Course in Probability Theory

20 1 MEASURE THEORY(A AB) A(C AD) = (A A C) A(B AD),AAB=C' A =BAC,AAB=CAD4 AAC=BAD .3 . If S2 has exactly n points, then J has 2" members . The B .F . generatedby n given sets "without relations among them" has 2 2 " members .4. If Q is countable, then J is generated by the singletons, andconversely . [HNT : All countable subsets of S2 and their complements forma B .F.]5 . The intersection of any collection of B .F .'s a E A) is the maximalB .F . contained in all of them ; it is indifferently denoted by I IaEA ~a or A aE a .*6 . The union of a countable collection of B .F .'s {5~ } such that :~Tj C z~+rneed not be a B .F ., but there is a minimal B .F . containing all of them, denotedby v j ; 7~ . In general V A denotes the minimal B .F. containing all Oa , a E A .[HINT : Q = the set of positive integers ; 9Xj = the B .F. generated by those upto j .]7. A B .F . is said to be countably generated iff it is generated by a countablecollection of sets . Prove that if each j is countably generated, then sois v~__ 1 .*8 . Let `T be a B .F. generated by an arbitrary collection of sets {E a , a EA) . Prove that for each E E ~T, there exists a countable subcollection {Eaj , j >1 } (depending on E) such that E belongs already to the B .F . generated by thissubcollection . [Hrwr : Consider the class of all sets with the asserted propertyand show that it is a B .F. containing each Ea .]9 . If :T is a B .F . generated by a countable collection of disjoint sets{A„ }, such that U„ A„ = Q, then each member of JT is just the union of acountable subcollection of these A,, 's .10 . Let 2 be a class of subsets of S2 having the closure property (iii) ;let .,/ be a class of sets containing Q as well as , and having the closureproperties (vi) and (x) . Then ~/ contains the B .F . generated by !7 . (This isDynkin's form of a monotone class theorem which is expedient for certainapplications . The proof proceeds as in Theorem 2 .1 .2 by replacing ~' 3 and -6-with 9 and .~/ respectively .)11 . Take Q = .:;4" or a separable metric space in Exercise 10 and let 7be the class of all open sets . Let ;XF be a class of real-valued functions on Qsatisfying the following conditions .(a) 1 E '' and l D E ~Y' for each D E 1 ;(b) dY' is 'a vector space, namely : if f i E f 2 E XK and c 1 , c 2 are anytwo real constants, then c1 f 1 + c2f2 E X' ;