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The winter term 1 exam for mathematics 420/507 real analysis i at the university of british columbia from the academic year 2007/2008. The exam covers topics such as the cartesian product, sigma algebra, borel measure, carathéodory extension theorem, regular borel measure, fatou and dominated convergence lemmas, lebesgue-radon-nikodym theorem, lebesgue differentiation theorem, l∞ norm, and properties of measurable functions.
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The University of British Columbia
Sessional Exams – 2007/2008 Winter Term 1 Mathematics 420/507 Real Analysis I Measure theory and Integration
Name:
Student Number:
This exam consists of 6 questions worth 100 marks in total. No aids are permitted.
Problem max score score
total 100
Do not define terms in theorems unless explicitly requested. If short of time, show good judgment by focusing on key steps.
(25 points) 1. (a)Define: the Cartesian product X =
α∈A E. What does the axiom of choice say about X?
(b) Define: σ algebra; Borel measure on Rn.
(c) State: the Carath´eodory extension theorem for an outer measure, defining terms in it that are not already defined.
(15 points) 3. (a)State the Fatou and Dominated Convergence Lemmas.
(b) Prove that limn→∞
−∞
n 1+(nx)^2 sin(x)^ dx^ exists and evaluate it. (First think about the graph of (^) 1+(nnx) 2 ).
(c) Let f be continuously differentiable. Prove that limn→∞
0 n
f (x + 1/n) − f (x)
dx exists and evaluate it. The mean value theorem may be useful.
(15 points) 4. (a)State the Lebesgue-Radon-Nikodym theorem.
(b) Let (X, M, μ) be a finite measure space, let N be a sub-σ-algebra of M, and let ν be the restriction of∫ μ to N. If f ∈ L^1 (μ) prove there exists g ∈ L^1 (ν) such that E f dμ^ =^
E g dν,^ ∀E^ ∈ N^.
(c) Give a counter-example to the conclusion of the previous part when μ is not finite.
(20 points) 6. (a)Define the L∞^ norm of a measurable function f defined on a measure space (X, M, μ).
(b) Let Ea = {x : |f (x)| ≤ a}. For a = ‖f ‖∞ prove that μ
Eac
= 0, and for all b < a, μ
Ebc
(c) Prove that ‖f + g‖∞ ≤ ‖f ‖∞ + ‖g‖∞ for f, g ∈ L∞.
(d) Let X be an uncountable set. Let M be the sigma algebra of sets E such that either E is countable or Ec^ is countable. Let μ be counting measure. Prove that if f ∈ L∞^ then (a) f is bounded by ‖f ‖∞ and (b) there exists a countable set E such that f is constant on Ec.
