MATH 303 Final, Fall 2001

Math 303 Fall 2002 Final Examination F Abi-Khuzam 1. Let p be a positive measure on X, (a) If fg are two non-negilive measurable functions satisfying f, dit < c.f gels < o, and [, fdu = f, gd for all measurable subsets of X. prove that f = g a.e. (b) If <7, } is a sequence of non-negative measurable functions on X, and f,, > f, and f, < f . prove that f fuel > | du. 2. Letfe L’(R) where | <p> &. a) Prove that lim). [, Wr +h) —Axldy = 0. ( (b) If fis as in part (a). g « LYCR) where g is the index conjugate to p, and F(x) = Jf t+ rg(ndt, prove that F is continuous on R. 3. Find each of the following limits and justify your answer. os a - ] (a) lim AUDEX oy, (b)limn? } (1 — x)" sinzxdy. oO crn ith A, Let fbe complex measurable on X and assume that w(X) < «. Let Alle = J pai ft = sup mite © X : Wool > 8), J ere If ¢ > 0. prove that i flle s Mpls i + HOO and conclude that lim,., || /» llo =O wheneverlim,.. |] fir ll = 0.