6.9 Establish the assertion in Remark 6.3.3 by completing the following steps: (a) Show that the coordinate map fr(r) = T, from R" to R is con- tinuous under the metric d of (3.3). (Conclude, using Example 1.1.6, that RN C B(Rx)). (b) Let C1 = {A : A = (a1, bi) x . . . x (ak, bk) X R", -00 B(R*). Let 9 be a o-algebra containing O3. Observe that given any open set A C R*, there exist a sequence of sets { Br}n21 in Of such that A = Un>, By (Problem 1.9). Since 9 is a o- algebra and B,, E G for all n 2 1, A e G. Thus, G is a o-algebra containing all open subsets of R", and hence G 3 B(R*). Hence, it follows that B(R) 70(01) 30(0)) = 0 938(R* ). 1.1 Classes of sets 13 Next note that any interval (a, b) C R can be expressed in terms of half spaces of the form (-co. r), re R as (a, b) = UI(-oo, b) \\ (-oo, a + n" )]. n=1 where for any two sets A and B, A\\B = (r : re Ar ( B). It is not difficult to show that this implies that o(0;) = B(R* ) for i = 2,4 (Problem 1.10)