MATH 303 Final, Fall 2003

Math 303-Final Exam (Fall 04) Problem 1 (25 pts) Suppose y is a measure on a set X, {f,} is a sequence of complex-valued measurable functions, and sup |fa(z}! < oc (x € X). If u(X) is finite, prove that to each ¢ > 0 there is a measurable set £ and a bound B,0< B< om, such that (i) u(B) <e (ii) |fa(z)| < B forn > land ce E°. In other words, show that off of sets of small measure, the sequence {/,} is uniformly bounded. What if u(X) is infinite ? Problem 2 (25 pts) Let p be a measure on a set X. If f is a complex-valued measurable function, put Ilfllico = sup tu({| fl > #}) <t<oo Illo= fez Let the complex-valued measurable functions f,, fo,... be such that lim {{fallieo = 0. N00 If yu is finite, prove that lim {| frllo = 0. n>0O Problem 3 (25 pts) Suppose 1 < p < oo and f € L?(R). (i) Show that, for z > 0, (41) Show that Problem 4 (25 pts) Let f : R + B® be a Lebesgue measurable function. Let & denote the set of all points z in R at which f is differentiable, ie. f(z@+h)~ fe) . h E={reR: lim exists}. ho0 Prove that £ is Lebesgue measurable.