This take-home midterm is due:

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1 Math 6110 Midterm 1 ( ) Name: Instr TA Gennady Uraltsev Yujia Zhai This take-home midterm is due: :59 If you end your training now if you choose the quick and easy path as Vader did you will become an agent of evil. This is a take-home exam. You are bound by the Cornell Code of Academic Integrity and agree to respect the rules below. This exam is individual. No collaboration is permitted. You are allowed to consult material listed on the Course Webpage and your personal notes. No external material may be consulted. The assignment must be turned in by the deadline via electronic means (scanned or compiled PDF) to guraltsev@math.cornell.edu or in hard copy to the mailbox of Gennady Uraltsev, Malott Hall, 5th floor. You may give for granted all material covered up to and including L09. For the purpose of solving any problem / part of a problem you may assume as true all the statements of preceding problems / parts e.g. you may assume the claims of 3.1 and 2.2 hold when solving 3.4. Problem Points Total (1) 4+3+3=10 (2) 5+10=15 (3) =20 (4) 6+4=10 (5) 4+4=8 (6) 9+7=16 (7) =21 Total: 100 Never tell me the odds!

2 Problem 1. Let (Ω, σ) be a space with a σ-algebra. We say that µ is a signed measure on (Ω, σ) if µ: σ R µ( ) = 0 µ( N E n ) = lim µ(e n ) for any pairwise disjoint (E n ) n N n N N n=0 and the series on the right converges absolutely. We say µ M 1 (Ω, σ) if µ is a signed measure and µ M 1 (Ω,σ) = sup µ(e) µ(ω \ E) <. E σ 1. Show that M 1 (Ω, σ) is a Banach space. 2. Show that if µ M 1 µ(a (Ω, σ) then µ (A) := sup E σ E) µ(a E c ) defines a positive finite measure. 3. Show that f µ f where µ f (E) := E f(x)dl(x) is an isometry of L1 (R, B(R), L) into M 1 (R, B(R)).

3 Problem 2. A functions F : R R is cadlag if x R lim F (y) exists y x y<x lim F (y) = F (x). y x y>x 1. Given a finite measure µ on (R, B(R)) let ( ) R µ (a) := µ (, a] R µ : R R +. Show that R µ is non-decreasing, cadlag and lim a R µ (a) = Show that given any non-decreasing cadlag function R: R R + such that lim a R(a) = 0 there exists a unique Borel measure µ such that R = R µ.

4 Problem 3. Recall the construction of the Cantor set. Let and C := N N C N. C N := {x [0, 1]: x = a n 3 n, a n {0, 1, 2}, a k 1 k N} n=1 1. Show that C N is Borel in [0, 1] and that L 1 (C) = Let D N (x) := L 1 (C N ) [0,x) 1 C N (x)dl 1 (x). Show that D N converge uniformly to a nondecreasing continuous function D : [0, 1] [0, 1] that is constant on any interval I [0, 1]\C. D is called the Devil s Staircase. 3. Using the results of Problem 2 show that there exists a measure µ: B([0, 1]) R + s.t. R µ = D and show that µ(c) = Show that µ({a}) = 0 for all a [0, 1] and that there is no f L 1 ([0, 1], B([0, 1]), L 1 ) s.t. µ(e) = f(x)1 E (x)dl 1 (x).

5 Problem 4. We want to show that given any Borel set A [0, 1] s.t. L 1 (A) > 0 there exists V A A that is not Lebesgue-Carathéodory measurable. Let V be the Vitali set such that q Q V + q = R and (V + q) (V + q ) = if q, q Q and q q. 1. Show that there exists q A Q s.t. either V A = (V + q A ) A is not measurable or L 1 (V A ) > Notice that V A + q [0, 2] q Q [0,1) and deduce that V A cannot be measurable and have L 1 (V A ) > 0

6 Problem 5. We want to show that continuous functions from R to R are not measurable if on the range one considers the σ algebra of Lebesgue-Carathéodory measurable sets. Recall that Lebesgue- Carathéodory measurable sets are the Carathéodory measurable sets of the Lebesgue outer measure and as such they contain Borel sets of R. 1. Let D : [0, 1] [0, 1] be the Devil s staircase. Show that F (x) = x + D(x) is a continuous strictly monotone surjective map of [0, 1] [0, 2] and that F (C) is Borel with L 1 (F (C)) = Let V F (C) F (C) be a non-measurable subset of F (C) (given by problem 4). Show that F 1 (V F (C) ) is Lebesgue-Carathéodory measurable but not Borel. Hint: Notice that F and F 1 are continuous and thus they send open sets into open sets.

7 Problem Show that translations are continuous on L p (R n, B(R n ), L n ) with p [1, ) i.e. lim f( x) f( ) L x 0 p (R n,b(r n ),L n ) = 0 f L p (R n, B(R n ), L n ). Hint: First show that the above statement holds if f Cc 0 (R n ). Cc 0 (R n ) L p (R n ) is dense in L p (R n ). Next use the fact that 2. Recall that K L p (R n, B(R n ), L n ) is compact if for any sequence (f n ) n N K there exists a subsequence (f kn ) n N such that lim n f kn f L p = 0 for some f K. Show that if K L p (R n, B(R n ), L n ) is compact then translations are uniformly continuous on K i.e. lim sup f( x) f( ) L x 0 p (R n,b(r n ),L n ) = 0 f K

8 Problem 7. Given f L 1 (R + \ {0}, B(R + ), L 1 ) let Hf be the function on R + \ {0} given by Hf(x) := 1 x 1. Show that Hf(x) is measurable. x 0 f(s)dl 1 (s). 2. Show that the operator f Hf is a linear map that satisfies the bounds Hf L (R + \{0},B(R + ),L 1 ) f L (R + \{0},B(R + ),L 1 ) Hf L 1, (R + \{0},B(R + ),L 1 ) f L 1 (R + \{0},B(R + ),L 1 ) 3. Show that for all p (1, ) there exists C p > 0 s.t. it holds that Hf L p (R + \{0},B(R + ),L 1 ) C p f L p (R + \{0},B(R + ),L 1 ) Hint: Use the (discretized) super-level set representation for the L p norm. Decompose f = k Z f k = k Z f(x)1 f [2 k,2 k+1 )(x). Notice that for any λ > 0 it holds that { {x: Hf(x) > 2 n ( } } x: H f k )(x) > 2n 1. k n 1 and using the L 1 L 1, bound on H one has ( ) µ {x: Hf(x) > 2 n } 2 n+1 f k L 1. k n 1