1 Al. (i) Let X be a set and SCP(X) a set of subsets of X. (a) Define what is meant by the o-algebra generated by S. (b)

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1 Al I Let X Be A Set And Scp X A Set Of Subsets Of X A Define What Is Meant By The O Algebra Generated By S B 1 (26.45 KiB) Viewed 34 times

1 Al. (i) Let X be a set and SCP(X) a set of subsets of X. (a) Define what is meant by the o-algebra generated by S. (b) Show that the set of all intervals with rational endpoints is a set of gener- ators for the o-algebra of Borel subsets B(R). (ii) Consider the interval (0.1) CR. Every x € (0,1) has a decimal expansion 1 = 0.222.23..., where x; € {0,1,...,9}. As in the course we will not allow the expansion to eventually end in repeated 9s to avoid ambiguity. Let B be the set of points in (0,1) whose decimal expansion contains infinitely many 5s. Show that B is a Borel set. 5