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 + 1)
n!(n k)
=
+
(k + 1)!(n k)! (k + 1)!(n k)!
(n + 1)!
n
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 integrability criterion to conclude that f is Riemann integrabl
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 positive integer.
(c) Let a and b be elements of a commuta
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 one of them has measure m/n.
2. Suppose that (fn )n=1,
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 sets A, B, and C.
2. Let A, B, C, and D be sets. Suppo
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, ) defined by (E) = 0 if E is countable and
(E) = 1 if
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 following statements:
(a) A closed subset of a complete metric
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 least upper bound in Q
Q.
(b) Let (rn )n=1,2. be the seque
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) If f (x) 6= 0 for each x X, then 1/f is a measurable fu
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 determine M .
Solution: By denition, () = 0 holds. If A, B R
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 : |F | n,
so that F(X) = Fn (X). As shown in class, the u
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 f (x) dx exists and equals
the Lebesgue integral [a,) f
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 have (A \ B) C = (A C) \ (B C).
Second, it follows from
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.
Proof . Let
f :=
n
X
Ej .
j=1
By our assumption, f (x) m f
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 countable and
(E) = 1 if E c is countable. Show that is
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 . Suppose that s Q
Q and s is an upper bound for the set
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 . Indeed, if x F , then there
exists r > 0 such that Br (x) F
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 ) (g), it is enough to prove the claim for f g. Since
|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 with
(A1 ) < , then
= lim (An ).
An
n
n=1
Does this remain
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 obtain that
sin x d(sin x) =
(sin x)(cos x) dx.
0
0
Su
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 this contradict the RadonNikodm Theorem?
y
Solution: As
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
quasi-integrable f , the function f A is also quasi-int
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, C S with A B C and (C \ A) = 0
Show that S is a -algeb