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 (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 (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
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
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
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 #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 #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 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