Question

Assume ECR is measurable throughout this problem set. #1) Let f : E [0, ∞) be measurable and assume that f is bounded, so that f(E) C [0, d) for some d>0. Here we see how to directly approximate f by an increasing sequence of sim- ple functions: For nЄ N and j = {0,1,...,2"} set y;= jd/2" and set Ij,n = [j-1, j) so that {n} partitions [0, d). Also, set Ej,n = f¹(In) so that {E}}=1 partitions E. If on is defined by 2" On(x)=-1XE₁n (x) j=1 where XE,,, is the characteristic function of Ej,n then prove that n→f uniformly on E and that for all x € E we have 41(x) ≤ 42(x) ≤ 43(x) ≤ ··· ≤ f(x). (In other words, {n(x)}=1 is an increasing sequence of nonnegative numbers.) Hint: Fix n = N and consider r = Ej,n. Look at two cases: f(x) is in the left half of Ij.n and f(x) is in the right half of Ij.n. Notice that these two halves are intervals in {I},n+1}}=-11 which allows you to explicitly compute n+1(x) depending on these two cases! #2) Here we prove a version of Egoroff's theorem for sets of infinite measure. Note that this is a slight generalization of August 2023 problem #7. Let E be measurable with m(E) = ∞. Let {f} be a sequence of measurable real valued functions on E where fn → f pointwise a.e. on E. =1 a) Suppose E' CE is measurable with m(E') < ∞. Show that we can write E' == UEk where each Ek is measurable, fnf uniformly on each Ek for k > 1 and m(E1) 0 Hint: By Egoroff's theorem, there exists measurable Ek for k > 2 where m(E\Ek) < 1/k and fnf uniformly on Ek. Now define E₁ in the obvious way... == b) Using a) show that we can write E=UEk where each Ek is measurable, fn → f uniformly on each Ek for k > 1 and m(E₁) = 0. Hint: First write E as the countable union of bounded, measurable sets like we have many times before!

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