Question: 4. (20 points) Let f : [0, 1] -> [0, co) be a Lebesgue measurable function. Assume that Jon f dm = 2020 2021 for

4. (20 points) Let f : [0, 1] -> [0, co) be a Lebesgue measurable function. Assume that Jon f" dm = 2020 2021 for n = 1, 2, 3, . .. , where m denotes the Lebesgue measure. 1. For k > 1, let Ek = f '([1 + #, +co)). Show that m(Ek) = 0. 2. Let E = f '((1, too)), show that m(E) = 0. 3. Show that f = YA a.c for some Lebesgue measurable set A C [0. 1)

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