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Math Foundations of Analysis 1. Let f € L¹ (R), and define x F(x) = / f(t)c dt. 8 Prove F is uniformly continuous. 2. Let f > 0 and f € I1 1(Rd. For a > 0, define Ex - {x € Rd:f(x)>a): = f 1 (a,00). Then prove that for every a > 0, we have < d.c. 3. Suppose f : Rd [0, x)] is measurable, and j f dx = C where 0 < C K 80. Let a be a constant. (a) Suppose a € (0,1). Prove that n-100 lim / n log 1 + n (Hint: Use Fatou's Lemma) (b) Suppose a € (1, xo). Prove that lim n da - 0. (Hint: Use DCT. See the final exam if you get stuck.) 4. Let 8 > 0, and for functions f : Rd R, define the dilation operator Ds by (a) Suppose f € Cc(Rd, the space of continuous functions with compact support. Prove that f is uniformly continuous. (b) Let f € Cc(Rd. Prove the following limit: lim 8-1+ (c) Now suppose f € L¹. Prove the following limit: lim IIDsf-fllp= - - 0. s-> 1 Instructor: Olakunle John- son Spring, 2017 (d) Don't write anything down for this part, but convince yourself the same argument (essentially) works to prove - = s Then with these two one-sided limits, we have for all f € L¹, , lim Il Dsf - fllor = 0. s->1

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