Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen March 29 2003

Weekly notice #9

The lectures in week 13: We went through all the details of the rather technical proof of Hovedsætning (Main Theorem) 5.4 and after that we proved the uniqueness of the Lebesgue measure in ℝ^{k}. As mentioned, the existence of the measure will not be proved. After this, a solution to Exercise W7.4 was presented. Moreover, it was proved that for any Riemann integrable function f :]a, b] → ℝ there exist two Borel functions s, t such that s ≤ f ≤ t and such that

∫_{]a,b]} s dm = ∫_{a}^{b} f(x) dx = ∫_{]a,b]} t dm.

In particular, s = t m-a.e.

The lectures in week 14: Portions of §5.2 and §5.3 will be covered. After this, the Cantor set Z (Example 5.22) will be (briefly) discussed. We will then discuss various measure theoretic paradoxes, and we will finish (hopefully) with Vitali’s Theorem (5.30).

Homework - to be handed in to the TA in week 15: Exercise 5.14 and 5.15. (See Exercise 5.8 for definitions.)

The problem sessions in week 15: W9.1, W9.2, W9.3, 5.7*, 5.8, 5.10, 5.11*, 5.32 (only 1^{∘}), 5.33*.

Problem W9.1. This is an exercise in σ-classes (Dynkin systems)!.

(i) Let 𝔻 be a σ-class on a set X. Suppose that A; B ∈ 𝔻 and that A ⊆ B. Show that B \ A ∈ 𝔻.

(ii) Let n ∈ ℕ, set X = {1, 2, …, n}, and consider the system of subsets

𝕂 = {{1}, {1, 2}, {1, 2, 3}, …, {1, 2, …, n − 1}, X, }.

Show that 𝕂(𝔻) = P(X). (iii)Consider here the system 𝕂 = {{1, 2}, {1, 3}} on X = {1, 2, 3, 4}. Show

that σ(𝕂) = P(X), and that 𝔻(𝕂) ≠ P(X), and conclude that 𝔻(𝕂) is not a σ-algebra.

(iv)Let X and 𝕂 be as in (iii). Construct two different measures, μ and ν, on P(X) such that μ(X) = ν(X) < ∞ and such that μ(K) = ν(K) for all K ∈ 𝕂.

Problem W9.2. Describe the Radon-measure on ℝ corresponding to each of the following positive linear forms:

I_{1}(f) = f(4), I_{2}(f) = f(1) + 2f(2), = I_{3}(f) = \frac{1}{2} ∫_{0}^{1} f(x) dx + f(0) + f(1),

where f ∈ C_{c}(ℝ). Problem W9.3. Let f : ℝ_{≥0} → ℝ be given by

f = ∑_{n=0}^{∞} \frac{(−1)^{n}}{n + 1}1_{[n,n+1[}.

Show that f ∈ L_{loc}(ℝ_{≥0}), that the limit

lim_{x→∞} ∫_{0}^{x} f(t) dt

exists, but that f does not belong to L(ℝ_{≥0}).