Exam June 2015, questions and answers PDF

Preview

Summary

Download Exam June 2015, questions and answers PDF

Description

Maths 730

THE UNIVERSITY OF AUCKLAND FIRST SEMESTER, 2015 Campus: City MATHEMATICS Measure Theory and Integration (Time allowed: THREE hours) NOTE: Please answer all 6 questions. There are 120 marks available. The questions are NOT of equal marks. The maximum number of marks, however, will be restricted to 100. Show all working and state clearly any theorems that you use in your answers.

–2– 1. (a) Give the definition of a σ-algebra.

Maths 730 (3 marks)

(b) Show that in general the union of two σ-algebras is not a σ-algebra. (Remark. If it is obvious that some set is a σ-algebra, then you do not need to prove this.)

(5 marks)

(c) Let X = IR and Σ = {{x} : x ∈ IR}. Show that A(Σ) 6= P(X).

(5 marks)

2. (a) Give the definition of an outer measure on a set X.

(3 marks)

∗

(b) Let µ be an outer measure on a set X. Let A, B ⊂ X and suppose that µ∗ (A∆B) = 0. Prove that µ∗ (A) = µ∗ (B). (Recall that A∆B = AB c ∪ Ac B.)

(6 marks)

(c) Consider IR with the Lebesgue σ-algebra and Lebesgue measure. Let A ⊂ IR be a set and suppose that the boundary ∂A is negligible. Prove that A is Lebesgue measurable. (Hint. A = A◦ ∪ (A ∩ ∂A).)

(7 marks)

3. Let (X, A, µ) be a measure space. Let f : X → [1, ∞) be a measurable function. Show that Z ∞ X f...