Richard Bamler Jeremy Leach office 382-N, phone: 650-723-2975 office 381-J [email protected] [email protected] office hours: office hours: Mon 1:00pm-3pm, Thu 1:00pm-2:00pm Thu 3:45pm-6:45pm

MATH 172: Lebesgue Integration and Fourier Analysis (winter 2012) Problem set 6

due Wed, 2/22 in class

(1) (20 points) Let X be an arbitrary space, M a σ-algebra on X and λ a measure on M. Consider an M-measurable function ρ : X → [0, ∞] which only takes non-negative values. (a) Show that the function, Z µ : M → [0, ∞],A 7→ (ρχA)dλ

is a measure on M. We will also denote this measure by µ = (ρλ). (b) Show that any M-measurable function f : X → R is integrable with respect to µ if and only if fρ is integrable with respect to λ. In this case we have Z Z Z fdµ = fd(ρλ) = fρdλ.

(Hint: Show first that this is true for simple functions, then for non-negative functions, then for general functions). (c) Show that if A ∈ M is a nullset with respect to λ, then it is also a nullset with respect to µ = ρλ. (Remark: A measure µ with this property is called absolutely continuous with respect to λ in symbols µ  λ.) (d) Assume that X = Rn, M = L and λ is the Lebesgue measure. Give an example for a measure µ0 on M such that there is no non-negative, M- measurable function ρ : X → [0, ∞] with µ0 = ρλ. (2) (20 points) This is a continuation from problem 3 on problem set 5. (a) Let f :[a, b] → R be a Riemann integrable function on the interval [a, b]. Show that f is integrable in the Lebesgue sense and that Z b Z f(x)dx = fdλ. a [a,b] (b) Let I ⊂ R be an interval (not necessarily closed or bounded) and f : I → R a function such that for any compact subinterval [a, b] ⊂ I, the restriction f|[a,b] is Riemann integrable. Assume moreover, that the function f itself is 1 2

integrable with respect to the Lebesgue measure. Show that the improper Riemann integral of f over I exists and is equal to the Lebesgue integral of f, i.e. Z Z bk Z f(x)dx = lim f(x)dx = fdλ k→∞ I ak I

for any sequence [a1, b1] ⊂ [a2, b2] ⊂ ... of closed intervals with I = S∞ k=1[ak, bk]. (c) Let I ⊂ R be again an arbitrary interval and let f : I → R be a function for which the improper Riemann integral exists and is finite but for which R R R one of the quantities I |f|dλ, I f+dλ, I f−dλ is infinite. Show that then all the other quantities are infinite. (d) Give an example for a continuous function f : (0, 1) → R for which the R 1 improper Riemann integral 0 f(x)dx exists and is finite, but which is not integrable with respect to the Lebesgue measure. (3) (10 points) (a) Give an example for a sequence of nonnegative continuous functions fk : R 1 [0, 1] → [0, ∞) such that limk→∞ fk(x) = 0 for all x ∈ [0, 1], but 0 fk(x)dx goes to ∞ as k → ∞. (k) (b) Use Fatou’s Lemma to show the following statement: Let (ai )i∈N be a sequence of sequences of non-negative real numbers, i.e. for every k ∈ N (k) (k) there is a sequence a1 , a2 ,.... Then ∞ ∞ X (k) X (k) lim inf ai ≤ lim inf ai . k→∞ k→∞ i=1 i=1 Give an example for which the inequality is strict. (c) Use Lebesgue’s Dominated Convergence Theorem to show the following (k) statement: Let (ai )i∈N be a sequence of sequences of (general) real numbers and assume that there is a sequence (bi)i∈N of non-negative real numbers (k) P∞ such that |ai | ≤ bi for all i, k ∈ N and i=1 bi < ∞. Moreover, assume (k) that limk→∞ ai exists of every i ∈ N. Then ∞ ∞ X (k) X (k) lim ai = lim ai . k→∞ k→∞ i=1 i=1 (4) (30 points) P∞ (a) Let a1, a2,... ∈ R be a sequence of real numbers such that k=1(ak)− = P∞ k=1(ak)+ = ∞, but |ak| → 0 as k → ∞. Show that for any z ∈ [−∞, ∞], 0 0 we can create a sequence a1, a2,... ∈ R by rearranging a1, a2,... such that ∞ X 0 ak = z. k=1 (Remark: By rearranging, we mean that there is a bijective map f : N → N 0 such that ak = af(k).) sin x (b) Consider the function f : R → R, f(x) = x . Prove that for every number z ∈ [−∞, ∞] there is a sequence of bounded Lebesgue sets Ak ⊂ R which 3 S∞ is increasing (i.e. A1 ⊂ A2 ⊂ ...) and which satisfies k=1 Ak = R such

that fχAk is integrable for all k and we have Z lim fχA = z. k→∞ k (c) Let f : R → [0, ∞) be an L-measurable function which takes only non- negative and finite values. Show that for any b > 0 there is a Lebesgue set B ⊂ R such that f is bounded on B and λ(B) > b. (d) Prove that the conclusion of (b) holds for any non-integrable, L-measurable R R function f : R → R for which f+dλ = f−dλ = ∞. (Hint: Use (a) and (c)). (e) Given any z ∈ R, show that there is a sequence fk : R → R of continuous sin x functions such that limk→∞ fk(x) = x for all x ∈ R and such that Z ∞ lim fk(x)dx = z. k→∞ ∞

Maximum total points: 80