Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen March 21 2003

Weekly notice #8

The lectures in week 12: This was more or less a guided tour through sections §4.3–4.6. Only a cursory knowledge of Sections §4.5–4.6 will be expected at the exam (you should be familiar with the material but not necessarily with the proofs).

The lectures in week 13: First it will be shown that the Riemann and Lebesgue integrals coincide when they are both defined (Problem W7.6). Then we will prove the uniqueness of the Lebesgue measure on ℝ^{k} (§5.1) and continue with §5.2 and portions of §5.3.

Homework - to be handed in to the TA in week 14: 3MI-exam January 2001: Problem 3.

The problem sessions in week 14: A) First do exercise 4.38 from the book. Then prove “Sætning 4.24” in the case J = ℕ. State “Sætning 4.26” and “Sætning 4.28” for this special case. Discuss when and how the content of Exercise 4.38 may be generalized to doubly indexed sequences {a_{j,k}} of complex numbers. [Hint: “Bemærkning 4.27”.] B) Do the following 3MI-exam problems: January ’01: Problem 2; June ’01: Problem 3; January ’02: Problems 3, 7. Finally, do W8.1.

[An English version of the above exams can be found below. For a Danish version, use e.g. the link “Old 3MI-exams (in Danish)” from the course home page.]

Problem W8.1. Consider the function f : ℝ → ℝ_{+} given by

f(x) = \frac{1}{1 + x^{4}} .

Let μ be the measure on 𝔹(ℝ given by

μ(E) = ∫_{E} f dm,

where m is the Lebesgue measure. Compute

∫_{0}^{∞} x dμ(x).

3MI-exam, January ’01 - Problem 2 (10 points): Determine the limit

lim_{n→∞} ∫_{−∞}^{∞} cos(\frac{x^{2} + 7n}{n})e^{−|x|} dx.

You must argue carefully for each step in the computation.

3MI-exam, January ’01 - Problem 3 (20 points): Let f : ℝ → ℝ be given by

f(x) = \{ \begin{matrix}1, & x ∈ [−1, 0[, \\ −2 & x ∈ [0, 4], \\ 0, & x ∈] − ∞, −1[ ∪ ]4, ∞[. \end{matrix}

Let m be the usual Lebesgue measure on ℝ, let ε_{0} be the Dirac measure in 0 on ℝ, and let μ denote the counting measure on ℝ.

(i) Argue that f is a Borel function.

(ii) Determine whether f ∈ L(m), determine whether f ∈ L(ε_{0}),

and determine whether f ∈ L(μ). (iii)Determine those of the integrals below, that are defined (see

question (ii))

∫_{ℝ} f dm, ∫_{ℝ} f dε_{0}, ∫_{ℝ} f dμ.

3MI-exam, June ’01 - Problem 3 (20 points): For every Borel set E ⊆ ℝ we define

μ(E) = ∫_{E} \frac{dm(x)}{1 + x^{2}},

where m as usual denotes the Lebesgue measure on ℝ.

(i) Justify (e.g. by reference to a relevant place in the notes) that μ is a measure on (ℝ, 𝔹(ℝ)) and that

∫ |x|^{a}μ(x) = ∫ \frac{|x|^{a}dm(x)}{1+x^{2}}  (⋆)

for every a ∈ ℝ. (As usual, 0^{a} = ∞ for a < 0, and |x|^{0} = 1 for all x in ℝ.)

(ii) Show that the value of (⋆) is finite when −1 < a < 1.

3MI-exam, January ’02 - Problem 3 (15 points): Set

F (t) = ∫_{0}^{π} sin(x + t cos(x))dx, t ∈ ℝ.

Justify that F is continuous and differentiable, and that F′(t) = ∫_{0}^{π} cos(x + t cos(x)) cos(x)dx, t ∈ ℝ.

3MI-exam, January ’02 - Problem 7 (25 points): Let (X, 𝔼, μ) be a measure space, let f : X → [0, ∞[ be an 𝔼-measurable function, and set

E_{t} = {x ∈ X | f(x) ≥ t}, t ∈ [0, ∞[.

a) Justify that E_{t} ∈ 𝔼 for t ∈ [0, ∞[.

Consider the function

φ(t) = μ(E_{t}), t ∈ [0, ∞[.

b) Show that φ is decreasing, that is, ∀s, t ∈ [0, ∞[: s < t ⇒ φ(s) ≥ φ(t).

c) Show that

lim_{n→∞} φ(1/n) = μ(X_{0}),

where X_{0} = {x ∈ X | f(x) > 0}.