Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen F ebruary 13 2003

Weekly notice #3

The lectures in week 7: §1 was finished and after that we started on §3 by introducing the concept of a measure μ on a σ-algebra 𝔼. We finished the lectures by introducing the following slightly non-standard terminology: A set N ∈ 𝔼 is a null-set if it has measure zero, i.e., if μ(N) = 0. The reason for this is that the Danish name, “nulmængde”, really is quite good. More generally, any subset (whether in 𝔼 or not) of a set N of measure zero, is called a null-set.

The problem sessions in week 9: 2.1, 2.3, 2.4, \underline{2.6}, \underline{W3.1}, W3.2, W3.3*.

(The underlined problems are the most important ones whereas those with a * are more exceptional and/or challenging and should be done only if time permits.)

Problem W3.1.

(i) What does the statement “f_{n} → f ε_{a}-a.e. (almost everywhere)” mean, when ε_{a} denotes the Dirac measure on P(X) located at a ∈ X, and f_{n} and f denote functions on X?

(ii) Same question as in (i) but with ε_{a} replaced by the counting measure on X.

(iii) Let f, g : ℝ → ℝ be continous functions and suppose that f = g m-a.e. (m is the Lebesgue measure). Prove that f = g everywhere.

(iv) Let the function g : ℝ → ℝ be given by: g(x) = 0 if x ≤ 0 and g(x) = 1 if x > 1. Is there a continuous function f : ℝ → ℝ such that f = g m-a.e.?

Problem W3.2. Let (X, d) be a metric space and let 𝔹 denote the Borel σ-algebra on X. Let B(X, ℂ) and C(X, ℂ) denote the sets of complex Borel functions and complex continuous functions on X, respectively. Explain why the following three facts hold true:

(i) If f, g ∈ B(X, ℂ) and λ ∈ ℂ then λ·f, f +g, and fg ∈ B(X, ℂ).

(ii) If (f_{n})_{n=1}^{∞} is a sequence in B(X, ℂ) and if the pointwise limit f(x) = lim_{n→∞} f_{n}(x) exists for all x ∈ X, then f ∈ B(X, ℂ).

(iii) C(X, ℂ) ⊆ B(X, ℂ).

Actually, B(X, ℂ) is the smallest family for which (i), (ii), and (iii) are true. Indeed,

Proposition W3. Let (X, d) be a metric space and let A be a family of complex functions on X such that:

(i) If f, g ∈ A and λ ∈ ℂ then λ · f, f + g, and fg ∈ A. (ii) If (f_{n})_{n=1}^{∞} is a sequence in A and if the pointwise limit f(x) =

lim_{n→∞} f_{n}(x) exists for all x ∈ X, then f ∈ A.

(iii) C(X, ℂ) ⊆ A. Then B(X, ℂ) ⊆ A.

You may begin to prove this proposition in Problem W3 below. The proof will be be completed in a problem on a later weekly notice.

Problem W3.3. Suppose A satisfies (i), (i), and (iii) in Proposition W3. Set 𝔼 = {A ⊆ X | χ_{A} = 1_{A} ∈ A}. Show that:

(i) 𝔼 is a σ-algebra. (ii) ∀V^{open} ⊆ X : V^{open} ∈ 𝔼.

(Hint: Look at the functions g_{n}(x) = min(n · dist(x, X \ V), 1)) (iii) 𝔹 ⊆ 𝔼.