Københavns Universitet

Examination in Mathematics 3MI

Solutions to selected questions (sketched).

Exercise 2 (15 points). Let \underline{a} = (a_{n})_{n=0}^{∞} be a complex sequence which belongs to ℓ_{4} = L_{4}(ℕ_{0}, P(ℕ_{0}), μ), where μ is the counting measure. Assume that ∥\underline{a}∥_{4}^{4} = 15^{3}. Prove that

∑_{n=0}^{∞} \frac{|a_{n}|}{2^{3·n}} ≤ 8.

Solution: Let \underline{b} = (b_{n})_{n=0}^{∞} be the complex sequence with b_{n} = \frac{1}{2^{3·n}}. Then \underline{b} ∈ ℓ_{4/3}, where 4/3 is the dual exponent to 4. A small computation shows that ∥\underline{b}∥ = (\frac{16}{15})^{3/4}, and the result now follows from the stated norm of \underline{a} in conjunction with Hölder’s Inequality. One must, however, also introduce the sequence \underline{|a|} = (|a_{n}|)_{n=0}^{∞} and observe that \underline{|a|} has the same ℓ_{4} norm as \underline{a}.

Exercise 3 (20 points). Let (X, 𝔼, μ) be a measure space. Let A, B, C ∈ 𝔼 satisfy

μ(A ∪ B) = 2, μ(A \ C) = \frac{3}{4}, μ(A) = \frac{3}{2}, μ(B) = \frac{1}{2}, μ(C) = 3.

(i) Determine μ(A ∩ B).

(ii) If, furthermore, μ(C \ (A ∪ B)) = 2, prove that μ(B ∩ C) = \frac{1}{4}.

(iii)Let B_{1} ⊇ B_{2} ⊇ ⋯ ⊇ B_{N} ⊇ … be a decreasing sequence of measurable sets, and assume ∀n : μ(B_{n} \ B_{n+1}) = 2^{−n}. Determine

μ (∪_{n=1}^{∞}(B_{n} \ B_{n+1})) .

(iv)Let C_{1} ⊇ C_{2} ⊇ ⋯ ⊇ C_{N} ⊇ … be a decreasing sequence of measurable sets, and assume that it now are the measures of the following set-differences; ∀n : μ(C_{n} \ C_{n+2}) = 2^{−n}, that are given, but where furthermore you are informed that μ(C_{1} \ C_{2}) = \frac{1}{4}. Determine

μ (∪_{n=1}^{∞}(C_{n} \ C_{n+1})) .

(v) Is the number computed in (iv) always equal to μ(C_{1})? - Explain!

Solutions to (iv) and (v): We have that

μ(C_{n} \ C_{n+2}) = μ(C_{n} \ C_{n+1}) + μ(C_{n+1} \ C_{n+2}) 1

which is the basic observation. Then,

∑_{n=1}^{∞} μ(C_{n} \ C_{n+2}) = ∑_{n=1}^{∞} \frac{1}{2^{n}} = 1

= ∑_{n=1}^{∞} μ(C_{n} \ C_{n+1}) + ∑_{n=1}^{∞} μ(C_{n+1} \ C_{n+2})

= ∑_{n=1}^{∞} μ(C_{n} \ C_{n+1}) + ∑_{n=1}^{∞} μ(C_{n} \ C_{n+1}) − μ(C_{1} \ C_{2}),

from which the result follows.

Another good idea is to use either the first or the second equality below:

∪_{n=1}^{∞}(C_{n}\C_{n+1}) = ∪_{n∈ℕ,n odd}^{∞}(C_{n}\C_{n+2}) = (C_{1}\C_{2})∪(∪_{n∈ℕ,n even}^{∞}(C_{n} \ C_{n+2})) .

Since

C_{1} = ∪_{n=1}^{∞}(C_{n} \ C_{n+1}) ∪ (∩_{n=1}^{∞}C_{n}) (disjoint union),

and we do not assume μ(∩_{n=1}^{∞}C_{n}) = 0, we cannot in general find the measure of C_{1} in this way.

Exercise 4 (15 points). Let ∀x ∈ ℝ : f(x) = e^{sin x}. Determine

lim_{n→∞} ∫_{−1}^{1} f(\frac{x}{n}) dx.

Exercise 5 (25 points). Let X, Y be non-empty sets and let ϕ : X ↦ Y be a function from X to Y.

(i) Let 𝔻_{X} be a Dynkin system (also called a σ-class) in X. Prove that

𝔻̃_{Y} = {F ∈ P(Y) | ϕ^{−1}(F) ∈ 𝔻_{X}}

is a Dynkin system in Y. (ii) Let 𝔼_{X} and 𝔼_{Y} be σ-algebras in X and Y, respectively, and set

𝔼̃_{Y} = {F ∈ 𝔼_{Y} | ϕ^{−1}(F) ∈ 𝔼_{X}}.

Prove that 𝔼̃_{Y} is a σ-algebra in Y.

(iii)Let 𝔼̂_{X} ⊆ P(X) be defined by

𝔼̂_{X} = {ϕ^{−1}(F) | F ∈ 𝔼_{Y} }.

Prove that 𝔼̂_{X} is a σ-algebra in X. (iv)Let 𝕂_{Y} be a generating system for 𝔼_{Y}. Prove that

𝕂̂_{X} = {ϕ^{−1}(K) | K ∈ 𝕂_{Y} }

is a generating system for 𝔼̂_{X}.

Solution to (iv): It is clear that 𝕂̂_{X} ⊆ 𝔼̂_{X}, hence it will generate a σ-algebra σ(𝕂̂_{X}) = 𝔼̌_{X} ⊆ 𝔼̂_{X}. If 𝔼̌_{X} ≠ 𝔼̂_{X} then

𝕂_{Y} ⊆ ˜ 𝔼̌_{Y} = {F ∈ 𝔼_{Y} | ϕ^{−1}(F) ∈ 𝔼̌_{X}} ≠ 𝔼_{Y} ,

and by (ii), this is a contradiction. Exercise 6 (15 points). a) Let g ∈ L_{2}(ℝ, 𝔹, dx), where dx denotes the Lebesgue measure on ℝ. Prove that the function

F (t) = ∫_{ℝ} \frac{1}{(x^{2} + t^{2} + 1)^{1/2}} g(x) dx

belongs to C^{1}(ℝ) and argue (or prove) that it actually belongs to C^{∞}(ℝ).

b) Let ψ ∈ S be a Schwartz function on ℝ. Prove that

∫ (∫ ψ̂(t)e^{−(x−t)^{2}/2} dt) dx = (\sqrt{2π})^{3}ψ(0),

where ψ̂ denotes the Fourier transform of ψ. You must carefully justify all steps in the computation.