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Math 447, Homework 7.∗ ∞ 1. Construct R a sequence of non-negative measurable functions {fn }n=1 such that fn → 0 uniformly on R, but fn = 1. Explain why neither the Monotone Convergence Theorem nor the Dominated Convergence Theorem apply to this example (i.e. verify that their assumptions are not satisfied). 2. Exercise 18.9 (page 320). 3. Exercise 18.22 (page 322). 4. Exercise 18.28 (page 327). Quiz 7. Let f ≥ 0 be integrable, and fix ε > 0. R R (a) Show that there exists a set E of finite measure such that E f > f − ε. (b) Show that E in part (a) can be chosen so that f is bounded on E. (c) Prove the following uniform R continuity result: there exists a δ > 0 such that for any set A with m(A) < δ, we have A f < ε. R (d) Conclude that the function F (x) = (−∞,x] f is continuous. ∗ c 2016 by Michael Anshelevich. 1 Math 447, Honors Homework 7 No difference this week.