(n) 4) Let E be measurable and f: E→ R be measurable. Let y=k/n for ke Z and ne N. Let Ik =)) so that clearly R = (). Let E() = f((n)) and define (n) 00 (n) On(x) = x()(x). k=-∞ Prove that each on is measurable and that on →f uniformly on R. Hint: = For the former, if on,N(x) Σk-N-1X(n) (x) then prove that we have lim No On,Non pointwise on E. Why is each on,N measurable? 4