Real Analysis II.

Instructor: Dmitry Ryabogin

Assignment II.

1. Problem 1. Suppose V is an open set in R^{k}, T maps V into R^{k}, and x ∈ V. The map T is called differentiable at x if there exists a linear operator A on R^{k} and a map α_{x} : V → R^{k} such that

T (x + h) − T (x) = Ah + α_{x}(h),  lim_{|h|→0} \frac{|α_{x}(h)|}{|h|} = 0.

Prove that A is unique.

Hint. Observe that

\frac{|α_{x}(λh)|}{|λh|}|h| → 0,

provided λ ∈ R, λ → 0.

2. Problem 2.

a) For n = 1, 2, 3, ... we let P_{n} be the set of all x ∈ R^{k} whose coordinates are integral multiples of 2^{−n}, and we let Ω_{n} be the collection of all 2^{−n} boxes with corners at points of P_{n}. Prove that every nonempty open set in R^{k} is a countable union of disjoint boxes belonging to Ω_{1} ∪ Ω_{2} ∪ Ω_{3} ∪ ....

b) Prove that for any positive translation-invariant Borel measure on R^{k} such that ν(K) is finite for every compact set K, there exists a constant c = c(ν) such that ν(E) = cμ(E) for all Borel sets E ⊂ R^{k}.

Hint. By a) it is enough to check b) for boxes from P_{n}. Let Q_{0} be a 1-box, put c = ν(Q_{0}). Then Q_{0} is a union of 2^{nk} disjoint 2^{−n} boxes that are translates of each other, and

2^{nk}ν(Q) = ν(Q_{0}) = cμ(Q_{0}) = c2^{nk}μ(Q)

for every 2^{−n} box.

c) Prove that to every linear transformation T of R^{k} into R^{k} corresponds a real number Δ(T) such that μ(T (E)) = Δ(T)μ(E) for every measurable E. In particular, μ(T (E)) = μ(E) when T is rotation.

Hint. Let m(E) = μ(T (E)). Prove that m is translation-invariant and use b).

d) Let T be a linear transformation on R^{k} defined as follows: T e_{1} = e_{1} + e_{2}, T e_{i} = e_{i} for i = 2, 3, ..., k. Assume also that Q is a cube consisting of all x = (x_{1}, ..., x_{k}) with 0 ≤ x_{k} < 1. Show that T (Q) is the set of all points ∑ x_{i}e_{i}, whose coordinates satisfy

x_{1} ≤ x_{2} < x_{1} + 1, 0 ≤ x_{i} < 1, i ≠ 2.

Show that Δ(T) = |detT |.

Hint. Let S_{1} be the set of points in T (Q) that have x_{2} < 1 and let S_{2} be the rest of T (Q). Observe that S_{1} ∪ (S_{2} − e_{2}) = Q, and that S_{1} ∩ (S_{2} − e_{2}) is empty. Hence, Δ(T) = 1.

3. Problem 3.

a) Let f ∈ L(R). Prove that for any ϵ > 0 there exists g_{ϵ} ∈ C(R) supported by a bounded interval, say [−A, A], such that

∫_{R} |f(x) − g(x)|dx < ϵ.

b) Let f ∈ L(R). Prove that for any ϵ > 0 there exists δ = δ(ϵ) such that

∫_{R} |f(x − t) − f(y − t)|dt < ϵ,

provided |x − y| < δ.

c) Prove that f ∗ g is continuous on the real line, provided f ∈ L(R) and g is bounded.

4. Problem 4. Suppose G is a subgroup of R (relative to addition), G ≠ R, and G is Lebesgue measurable. Prove that μ(G) = 0.

Hint. Use the last problem of the previous assignment.

5. Problem 5^{∗}. Prove that the Cantor function is log_{3} 2-Lipschitz, i.e., there exists a constant K such that for all x, y ∈ [0, 1],

|c(x) − c(y)| ≤ K|x − y|^{log_{3} 2}.