Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen F ebruary 5 2003

Weekly notice #2

The lectures in week 6: The first hour was an introduction to the course. The Riemann integral was discussed, and an overwiew of the Lebesgue integral was offered. In the second hour, some time was spent on §0 and the rest on §1. The presentation of §0 was rather sketchy and the details are left to yourselves to fill in. We will return to §1 in week 7 and fill in some details. After this we move on to §3 and first after that to §2. Written assignment - to be handed in in week 8 and graded by the teaching assistant: 1.7 and 3.1 The problem sessions in week 8: The three problems below on this page together with 3.3^{∗}, 3.4^{∗}, 3.5, 3.8^{∗}, 3.9^{∗}, 3.12, 3.13.

The * indicates a problem which should be done if time permits..

Problem 2.1. Show that each of the three sets

ℚ ,  ⋃_{n=1}^{∞} [n, n + \frac{1}{n^{2}}] ,  ⋃_{n=1}^{∞}[n, n + \frac{4}{n^{2}}]

are Borel sets. Compute the Lebesgue measure of each one.

Problem 2.2. Show that m(U) > 0 for any open non-empty subset of ℝ, where m = m_{1} denotes the Lebesgue measure on ℝ. Proceed to ℝ^{k} and prove the analogous statement.

Problem 2.3. Let ℚ ∩ ]0, 1] = {q_{1}, q_{2}, q_{3}, … } (this is, after all, a denumerable set!) For each ε > 0 let

U_{ε} =]0, 1[ ⋂ (⋃_{n=1}^{∞} ]q_{n} − ε · 2^{−n−1}, q_{n} + ε · 2^{−n−1}[) .

Prove that U_{ε} is open, that m(U_{ε}) ≤ ε, and that \overline{U_{ε}} = [0, 1].

Set F_{ε} = [0, 1] \ U_{ε}. Prove that F_{ε} is closed, m(F_{ε}) ≥ 1 − ε, and that int(F_{ε}) = F_{ε}^{o} = ∅.