Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen April 27 2003

Weekly notice #12

The lectures in week 16: We covered product measures and Tonelli’s Theorem and finished by actually computing an integral (of e^{−x^{2}}, over ℝ).

The lectures in week 18: Some discussion of Fubini’s Theorem and applications, but proofs will be left to yourselves. After that we embark on §7: L_{p} spaces, where we ought to reach at least Hölders Inequality.

Notice: Exam Exercises may be found (in Danish) via the course home page. In a few days, they will appear here also.

Homework - to be handed in to the TA in week 19: 3MI-Exams: January ’02 Exercise 6 and May ’01 Exercise 5. The problem sessions in week 19: W12.1, 6.22, 6.25, 6.26; 3MI-Exam June 1999: Exercises 4, 5*; 7.1*

Problem W12.1. Let ε_{2} be the Dirac measure on ℝ, given on any subset A of ℝ by: ε_{2}(A) = 1 if 2 ∈ A, and 0 otherwise. Let G ⊆ ℝ^{2} be given by

G = {(x, y) ∈ ℝ^{2} | x ≥ 0, y ≥ 0, x + y ≤ 4}.

Determine (m ⊗ ε_{2})(G). Let f : ℝ^{2} ↦ ℂ be given by f(x, y) = (x^{2} + y^{2} + 1)^{−1}. Compute

∫_{ℝ^{2}} f d(m ⊗ ε_{2}).

Set μ = m ⊗ ε_{2} + ε_{2} ⊗ m. Compute

∫_{ℝ^{2}} f dμ.