Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen April 14 2003

Weekly notice #11

The lectures in week 15: We discussed the Cantor set Z. After this, §6.1 was completed.

The lectures in week 16: Product measures. Theorems of Fubini and Tonelli.

Homework - to be handed in to the TA in week 18: 3MI-Exam January ’00; Exercises 2 and 4 (see below)

The problem sessions in week 18: W11.1, W11.2, W11.3, 6.14, 6.15, 6.17 (you may here assume that X and Y are countable), 6.18, 6.19, 6.20.

Problem W11.1: Let (X, 𝔼, μ) be a σ-finite measure space and let f : X ×X → ℂ be a measurable function that satisfies f(x, y) = −f(y, x). Show that if f ∈ L(μ ⊗ μ) then

∫_{X} (∫_{X} f(x, y) dμ(x)) dμ(y) = 0.

Give an example of a function f which satisfies f(x, y) = −f(y, x) and for which the integral above is defined and ≠ 0. [Hint: Consider a suitable rational function q(x, y)in two variables in [0, 1)×[0, 1], i.e., q(x, y) = p_{1}(x, y)/p_{2}(x, y) with p_{1}, p_{2} polynomials.]

Problem W11.2: Let I_{p}, I_{q} be standard intervals in ℝ^{p} and ℝ^{q}, respectively. Let f be a real Borel function on ℝ^{k}, where k = p + q. Show that if

∫_{I_{q}} ∫_{I_{p}} |f(x, y)| dx dy < ∞

then both double integrals

∫_{I_{q}} ∫_{I_{p}} f(x, y) dx dy and ∫_{I_{p}} ∫_{I_{q}} f(x, y) dy dx

exist and are equal.

Problem W11.3: (Addendum to exercise 6.14) Let (X, 𝔼, μ) be a σ-finite measure space and let f ∈ M^{+}(X, 𝔼) (with finite values). Show that

∫_{X} f dμ = ∫_{0}^{∞} μ(f^{−1}(]t, ∞[)) dt.

[Hint: Look at the product space (X × ℝ, 𝔼 ⊗ 𝔹, μ ⊗ m) and at the sets

(1)  G(f) = {(x, t) | 0 ≤ t < f(x)} ⊆ X × ℝ.

Don’t forget to skow that G(f) ∈ 𝔼 ⊗ 𝔹. This may be done by first looking at the case where f is simple and then by using that G(f) = ∪_{n=1}^{∞}G(s_{n}) if s_{n} ↗ f.]

Remark: The function t ↦ μ(f^{−1}(]t, ∞[)) is decreasing and hence, actually, the generalized Riemann integral exists. One can thus define the Lebesgue integral w.r.t. an arbitrary (σ-finite) measure μ by (1) above.

January 2000 Exercise 2: Let E be the set of points (x, y) ∈ ℝ^{2} for which either x or y is rational. Show that E is a Borel subset of ℝ^{2} and determine m_{2}(E), where m_{2} denotes the Lebesgue measure on ℝ^{2}.

January 2000 Exercise 4: Set A = {(x, y) ∈ ℝ^{2} | 0 < x ≤ y ≤ π/2} ⊆ ℝ^{2},

and let f : ℝ^{2} → ℝ be given by

f(x, y) = \{ \begin{matrix}cos(y)/y & if (x, y) ∈ A \\ 0 & if (x, y) ∈ ℝ^{2} \ A \end{matrix}

Carefully explain why

∫_{ℝ^{2}} f(x, y) dm_{2}(x, y) = ∫_{ℝ} (∫_{ℝ} f(x, y) dm(x)) dm(y) = ∫_{ℝ} (∫_{ℝ} f(x, y) dm(y)) dm(x),

and then compute the integral.