MATH 417
Assignment #2
1. For k, n IN0 with k n, we define nk := n!/(k!(n k)!).
n
n
(a) Show that n+1
k+1 = k + k+1 for k < n.
Proof . We have
n
n
n!
n!
+
=
+
k
k+1
k!(n k)! (k + 1)!(n k 1)!
n!(k +
MATH 417
Assignment #2
due on Friday, September 25, 2009
1. (20 points) For k, n IN0 with k n, we define nk := n!/(k!(n k)!).
n
n
(a) Show that n+1
=
+
k+1
k
k+1 for k < n.
n
(b) Prove that k is a pos
MATH 417
Assignment #7
due on Monday, November 30, 2009
1. Let E1 , E2 , . . . , En be Lebesgue measurable subsets of [0, 1]. If each point of [0, 1]
belongs to at least m of these sets, then at least
MATH 417
Assignment #1
due on Wednesday, September 16, 2009
1. The symmetric difference of two sets A and B is defined to be the set
AB := (A \ B) (B \ A).
Show that (AB)C = A(BC) for every triplet of
MATH 417
Assignment #5
due on Monday, November 9, 2009
1. Let X be an uncountable set, and let
S := cfw_E X : E or E c is countable.
(a) Show that S is a -algebra.
(b) Let be the function from S to [0
MATH 417
Assignment #4
due on Friday, October 23, 2009
1. For subsets A and B of a metric space (X, ) show that
(a) A B = A B.
(b) A B A B.
(c) A B A B, provided B is an open set.
2. Prove the followi
MATH 417
Assignment #3
due on Friday, October 9, 2009
1. The following problems are concerned with incompleteness of the rational numbers.
(a) Let A := cfw_r Q
Q : r2 2. Show that the set A has no lea
MATH 417
Assignment #6
due on Friday, November 20, 2009
1. Let (X, S) be a measurable space, and let f : X IR be a measurable function.
(a) Show that |f |p is a measurable function for all p > 0.
(b)
Math 417 (Fall 2013)
Honors Real Variables, I
Final Practice Problems (with Solutions)
1. Dene
0,
if A = ,
A
1, if A = is bounded,
,
otherwise.
: P(R) [0, ],
Show that is an outer measure, and deter
Math 417 (Fall 2012)
Honors Real Variables, I
Solution #3
1. Let f : [0, 1] R be dened as follows:
f (t) :=
0, t = 0 or t Q,
/
1
n
m , t = m with n, m N such that gcd(n, m) = 1.
Use Lebesgues integrab
Math 417 (Fall 2013)
Honors Real Variables, I
Solutions #1
1. Let X be a countable set. Show that
F(X) := cfw_F X : F is nite
is a countable subset of P(X).
Solution: For n N, set
Fn (X) := cfw_F X :
Math 417 (Fall 2013)
Honors Real Variables, I
Solutions #5
1. Let f : [a, ) R be Lebesgue integrable such that f |[a,b] is Riemann integrable for
each b > a. Show that the improper Riemann integral a
MATH 417
Assignment #1
1. The symmetric difference of two sets A and B is defined to be the set
AB := (A \ B) (B \ A).
Show that (AB)C = A(BC) for every triplet of sets A, B, and C.
Proof . First, we
MATH 417
Assignment #7
1. Let E1 , E2 , . . . , En be Lebesgue measurable subsets of [0, 1]. If each point of [0, 1]
belongs to at least m of these sets, then at least one of them has measure m/n.
Pro
MATH 417
Assignment #5
1. Let X be an uncountable set, and let S := cfw_E X : E or E c is countable.
(a) Show that S is a -algebra.
(b) Let be the function from S to [0, ) defined by (E) = 0 if E is c
MATH 417
Assignment #3
1. The following problems are concerned with incompleteness of the rational numbers.
(a) Let A := cfw_r Q
Q : r2 2. Show that the set A has no least upper bound in Q
Q.
Proof .
MATH 417
Assignment #4
1. For subsets A and B of a metric space (X, ) show that
(a) A B = A B.
(b) A B A B.
(c) A B A B, provided B is an open set.
c
Proof . We first show that E F implies E F . Indee
Math 417 (Fall 2012)
Honors Real Variables, I
Solutions #4
1. Let (X, S, ) be a measure space, and let f, g : X [, ] be -integrable. Show
that also f g and f g are -integrable.
Solution: As f g = (f )
Math 417 (Fall 2013)
Honors Real Variables, I
Solutions #2
1. Let X be a set, let S be a -algebra over X, and let : S [0, ] be a measure.
Show that, whenever A1 A2 A3 is a decreasing sequence in S wit
Math 417 (Fall 2013)
Honors Real Variables, I
Solutions #8
1. Evaluate the LebesgueStieltjes integrals
3
sin x d(sin x)
and
0
x2
1
1
d(x2 ).
+ 4x 3
Solution: Since sin is continuously dierentiable, we
Math 417 (Fall 2013)
Honors Real Variables, I
Solutions #7
1. Let : P(R) [0, ] be counting measure. Show that
function g : R [0, ] such that
, but that there is no
g d
(A) =
A
for A B(R). Why doesnt
Math 417 (Fall 2013)
Honors Real Variables, I
Solutions #6
1. Let (X, S, ) be a measure space. We call a function f : X [, ] quasiintegrable if f + < or f < , in which case we set f := f + f . As for
Math 417 (Fall 2013)
Honors Real Variables, I
Midterm Practice Problems (with Solutions)
1. Let (X, S, ) be a measure space. The completion of S with respect to is dened
as
S := cfw_B X : there are A,