PROBLEM SET 2a

1. Prove that the Lebesgue function F satisfies

|F (x) − F (y)| ≤ |x − y|^{α},

where α = log 2/ log 3. This is called a Hölder condition of order α.

2. If E ⊂ [0, 1] is a null set, and f : [0, 1] → [0, 1] is continuous, does f(E) have to be a null set as well? Prove this or find a counterexample.

3. Let X = {x + y : x, y ∈ C} be the set of sums of numbers in the Cantor middle third set. What is X?

4. Prove that if λ > 0 then m^{∗}(λE) = λm^{∗}(E) where λE = {λx : x ∈ E}.

5. If X is set of finite Lebesgue measure show that m(X ∩ X + t) → 0 as t → ∞. Here X +t = {x+t : x ∈ X}. Does there have to be a value of t so that m(X ∩X +t) = 0?