350 CHAPTER 14. THE LEBESGUE INTEGRAL

15. Lebesgue measure was discussed. Recall that m((a,b)) = b− a and it is definedon a σ algebra which contains the Borel sets. Show, using Dynkin’s lemma, thatm is translation invariant meaning that m(E) = m(E +a). for all Borel sets E andthen explain why this will be true for all Lebesgue measurable sets described inProblem 14. Let x∼ y if and only if x− y ∈Q. Show this is an equivalence relation.Now let W be a set of positive measure which is contained in (0,1). For x ∈W,let [x] denote those y ∈W such that x ∼ y. Thus the equivalence classes partitionW . Use axiom of choice to obtain a set S ⊆W such that S consists of exactly oneelement from each equivalence class. Let T denote the rational numbers in [−1,1].Consider T+S ⊆ [−1,2]. Explain why T+S ⊇W . For T≡

{r j}, explain why the

sets{

r j +S}

j are disjoint. Explain why S cannot be measurable. Explain why thisshows that for any Lebesgue measurable set of positive measure, there is a subset ofthis set which is not measurable.

16. Consider the sequence of functions defined in the following way. Let f1 (x) = x on[0,1]. To get from fn to fn+1, let fn+1 = fn on all intervals where fn is constant. Iffn is nonconstant on [a,b], let fn+1(a) = fn(a), fn+1(b) = fn(b), fn+1 is piecewiselinear and equal to 1

2 ( fn(a)+ fn(b)) on the middle third of [a,b]. Sketch a few ofthese and you will see the pattern. The process of modifying a nonconstant sectionof the graph of this function is illustrated in the following picture.

Show { fn} converges uniformly on [0,1]. If f (x) = limn→∞ fn(x), show that f (0) =0, f (1) = 1, f is continuous, and f ′(x) = 0 for all x /∈ P where P is the Cantor setof Problem 14. This function is called the Cantor function.It is a very importantexample to remember. Note it has derivative equal to zero a.e. and yet it succeeds inclimbing from 0 to 1. Explain why this function cannot be recovered as an integralof its derivative even though the derivative exists everywhere but on a set of measurezero. Hint: This isn’t too hard if you focus on getting a careful estimate on thedifference between two successive functions in the list considering only a typicalsmall interval in which the change takes place. The above picture should be helpful.

17. ↑ This problem gives a very interesting example found in the book by McShane [23].Let g(x) = x+ f (x) where f is the strange function of Problem 16. Let P be theCantor set of Problem 14. Let [0,1] \P = ∪∞

j=1I j where I j is open and I j ∩ Ik = /0if j ̸= k. These intervals are the connected components of the complement of theCantor set. Show m(g(I j)) = m(I j) so

m(g(∪∞j=1I j)) =

∞

∑j=1

m(g(I j)) =∞

∑j=1

m(I j) = 1.

Thus m(g(P)) = 1 because g([0,1]) = [0,2]. By Problem 15 there exists a set,A⊆ g(P) which is non measurable. Define φ(x) = XA(g(x)). Thus φ(x) = 0 unlessx ∈ P. Tell why φ is measurable. (Recall m(P) = 0 and Lebesgue measure is com-plete.) Now show that XA(y) = φ(g−1(y)) for y ∈ [0,2]. Tell why g−1 is continuousbut φ ◦ g−1 is not measurable. (This is an example of measurable ◦ continuous ̸=