MATHEMATICS 580
HOMEWORK 2
SOLUTIONS
1. Dene the function F (x) = d(x, f (x). Then given x, y K we have
F (x) = d(x, f (x) d(x, y ) + d(y, f (y ) + d(f (y ), f (x)
2d(x, y ) + d(y, f (y ) = 2d(x, y ) + F (y ).
So F (x) F (y ) 2d(x, y ). By interchanging
Mathematics 580: Real Analysis
Midterm Review
I. Denitions and statements of theorems.
1. Given a sequence, cfw_ xn in R, give two equivalent, yet dierent, definitions of lim supn xn and lim inf n xn .
lim sup xn = lim sup xk = supcfw_ x : cfw_ xnk with
Mathematics 580: Real Analysis
Midterm Review
I. Denitions and statements of theorems.
1. Given a sequence, cfw_ xn in R, give two equivalent, yet dierent, definitions of lim supn xn and lim inf n xn .
2. Given a countable family of sets cfw_En , give tw
MATHEMATICS 580
FINAL EXAM
DUE THURSDAY, DECEMBER 12
Please deliver your paper by 5:00 PM Thursday December 12 to my oce in 149
Gordon Palmer hall. If I am not there you may slide it under my door. In working
the exam, you may consult me, your lecture not
MATHEMATICS 580: REAL ANALYSIS I
MIDTERM, OCTOBER 9, 2013
I. Theory
1. Let (X, d) be a metric space.
(a) What is the Borel -algebra, BX ?
The Borel -algebra is the smallest -algebra that contains all the open sets in X .
(b) Give three dierent families of
MATHEMATICS 580
HOMEWORK 1 SOLUTIONS
DUE SEPTEMBER 4
1. We will show that the sequence cfw_xn is Cauchy, hence convergent since X is
complete. Since the series converges, given > 0 we can nd N such that m
n N implies m n d(xk , xk+1 ) < . By the triangl
MATHEMATICS 580
FINAL EXAM
DUE THURSDAY, DECEMBER 12
Please deliver your paper by 5:00 PM Thursday December 12 to my oce in 149
Gordon Palmer hall. If I am not there you may slide it under my door. In working
the exam, you may consult me, your lecture not
MATHEMATICS 580
HOMEWORK 5
SOLUTIONS
1. The point of this exercise is to show that the measurability assumption of the
function f (x, y ) on the product space is important. Consider the measure spaces
(X, M, ) = (Y, N , ) = ([0, 1], L, m).
Notice both of
MATHEMATICS 580
HOMEWORK 4
SOLUTIONS
1. Let f be the Cantor function and dene (x) = x + f (x).
(a) Prove that is a continuous bijection from [0, 1] to [0, 2], and therefore has
an inverse function 1 : [0, 2] [0, 1] that is continuous. (Hint: Show that
is
MATHEMATICS 580
HOMEWORK 3
SOLUTIONS
1. Suppose (X, M) is a measurable space, g : X R is measurable, and f : R R
is Borel measurable cfw_x R : f (x) > a BR for all a R. Prove that h = f g
is measurable. (Hint: use exercise 4 (b) on homework assignment 2.)