MATH 467/MAST 669/837 Measure Theory
Solutions to Assignment #1
1. (Ex. 6, page 34) Let A be the set of irrational numbers in the interval [0, 1]. Prove that m (A) = 1.
Proof. Let B = Q [0, 1], the set of rational numbers in [0, 1], and note that B is cou
Math 507/420: Measure Theory and Integration (2010) SOLUTIONS
Homework Assignment #1
Due: Friday, Sept. 24, at beginning of class.
You may use any result from Chapter 0 or Sections 1.1., 1.2. or 1.3 of Folland or established in class.
1. True or False (ju
T. Liggett
Mathematics 131C Midterm Solutions
May 5, 2010
(25) 1. Consider solving the equations
u3 + xv y = 0
v 3 + yu x = 0
(1)
for u, v in terms of x, y .
(a) Show that there are dierentiable functions u(x, y ) and v (x, y ) dened
in a neighborhood N o
T. Liggett
Mathematics 131C Final Exam Solutions
June 7, 2010
(25) 1. (a) State Fatous Lemma.
See Royden, page 86.
(b) State the Bounded Convergence Theorem.
See Royden, page 84.
(c) Use Fatous Lemma to prove the Bounded Convergence Theorem.
Suppose |fn |