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AMAT 510A Spring 2024 HW 6 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 n = N and j = {0, 1, ..., 2n} set y; = jd/2" and set Ij,n = [Yj−1, Yj) so that {Ij,n}}₁ partitions [0, d). Also, set Ej,n = ƒ¯¹(Ij,n) so that {Ej,n}}¾½-1 partitions E. 1 If On is defined by շո Yn(x) = ΣYj−1XEj,n(x) j=1 where XEj,n is the characteristic function of Ej,n then prove that n→ f uniformly on E and that for all x Є E we have 41(x) ≤ 42(x) ≤ 43(x) ≤ …… · ≤ f(x). (In other words, {n(x)}x±1 is an increasing sequence of nonnegative numbers.) Hint: Fix n ≤ N and consider x Є 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 {Ij,n+1}+1 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 {ƒn}x_₁ be a sequence of measurable real valued functions on E where ƒn → ƒ pointwise a.e. on E. • :1 = a) Suppose E' CE is measurable with m(E') < ∞. Show that we can write E' U1Ek where each Ek is measurable, fn → f 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 ƒn → ƒ uniformly on Ek. Now define E₁ in the obvious way... =1 • b) Using a) show that we can write E = U. Ek where each Ek is measurable, fn → ƒ 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! 1 #3) Let E be measurable and f : E → R be measurable. If ƒ satisfies the property that ACE and m(A) = 0 implies m(f(A)) = 0 then prove that F(E) can be written as the union of a Fo set and a set of measure zero. Note: this is a slight generalization of January 2023 Problem #7. Hint: This is similar to Problem 2): write E as the countable union of bounded, mea- surable sets like we have many times before. On each of these sets, use Lusin's theorem in a way similar to that of Problem 26). Remember that if g: R → R is continuous and CCR is compact then g(C) is also compact. #4) a) Suppose that P(x) is a statement for each x = R such that P(x) is true for a.e. x Є R. If E C R then also prove that P(x) is true for a.e. x = E (that is, prove that m({x = E : P(x) is false }) = 0. = b) Let f E → R be measurable. Prove that there exists continuous functions gn: R→ R such that limn→∞ In Hint: f pointwise a.e. on E. Like in last HW, use the Borel Cantelli Lemma, but this time in conjunction with Lusin's theorem and a)! Note: this is a generalization of August 2017, Problem #4. #5) Let : E → [0, ∞) be a simple function, so that (E) nonnegative numbers y1,. ...., Yn. Suppose that we also have m φ(α) = ΣαΧΑ; (2) i=1 = {1,..., Yn} for distinct for pairwise disjoint subsets A¿ of E and nonnegative numbers a₁,..., am. For 1 ≤ j ≤ n let I; = {i = {1, ..., m} : ai = y; }. a) Prove that n {1,...,m} = || Ij. j=1 (In other words, {1, ..., m} is the disjoint union of the I;'s) • b) If we also assume that E = || 1 A₁ then prove that for each j = {1,..., n} we have E; = | | Ai iЄIj where E; = 篹({Yj}). 2