Richard Bamler
office 382-N, phone: 650-723-2975
[email protected]
office hours:
Mon 1:00pm-3pm, Thu 1:00pm-2:00pm
Jeremy Leach
office 381-J
[email protected]
office hours:
MonWedFri 2:15pm-4:15pm
MATH 172: Lebesgue Integration and Fourier Analysis
(winter 2012)
Problem set 1
due Wed, 1/18
(1) (20 points)
Here we complete an argument from class: Assume that there is
a function vol : P(Rn ) → [0, ∞] which satisfies the properties
(ii) invariance under Euclidean transformations: If A ⊂ Rn and Φ : Rn → Rn
is a Euclidean transformation, then vol(Φ(A)) = vol(A).
(iii) normalization: vol([0, 1]n ) = 1.
(iv) σ-additivity: If A1 , A2 , . . . ⊂ Rn are pairwise disjoint, then
vol(A1 ∪ A2 ∪ . . .) = vol(A1 ) + vol(A2 ) + . . . .
(Observe that we use the convention x + ∞ = ∞, ∞ + ∞ = ∞.) As we have
seen in class, such a function does actually not exist, this is part of the proof by
contradiction!
(a) Show that (under the assumptions above) vol(∅) = 0.
(b) Show that (ii)-(iv) imply
(i) finite additivity: If A1 , A2 , . . . , Ak ⊂ Rn are pairwise disjoint, then
vol(A1 ∪ A2 ∪ . . . ∪ Ak ) = vol(A1 ) + vol(A2 ) + . . . + vol(Ak ).
(c) Show that if A ⊂ B ⊂ Rn , then vol(A) ≤ vol(B).
(d) Show that if A, B ⊂ Rn , then vol(A ∪ B) = vol(A) + vol(B) − vol(A ∩ B).
(e) Show that if A1 , . . . , Ak ⊂ Rn , then vol(A1 ∪ . . . ∪ Ak ) ≤ vol(A1 ) + . . . +
vol(Ak ).
(f) Show that for any numbers a1 , . . . , an , b1 , . . . , bn ∈ R with ai < bi and
bi − ai = m1i for some natural numbers m1 , . . . , mn ∈ N, we have
(b1 − a1 ) · · · (bn − an ) ≤ vol([a1 , b1 ] × . . . × [an , bn ])
≤ 2n (b1 − a1 ) · · · (bn − an ).
1
2
(g) Show that for any numbers a1 , . . . , an , b1 , . . . , bn ∈ R with ai < bi and
bi − ai = 2m1 i for some nonnegative integers m1 , . . . , mn ∈ N0 , we have
vol([a1 , b1 ] × . . . × [an , bn ]) = (b1 − a1 ) · · · (bn − an ).
(Hint: use (d) and estimate the volume of the intersection by small rectangles, then apply (f))
(h) Show the same as in (g), but under the assumption that bi − ai = 2pmi for
i
p1 , . . . , pn ∈ N and m1 , . . . , mn ∈ N0 .
(i) Finally, show that for any real numbers bi ≥ ai we have
vol([a1 , b1 ] × . . . × [an , bn ]) = (b1 − a1 ) · · · (bn − an ).
(Hint: use approximation.)
(2) (10 points)
Prove De Morgan’s Law: For any subsets Aι ⊂ Rn (ι ∈ I,
where I is some index set), we have
[ c \
\ c [
c
Aι =
Aι
and
Aι =
Acι .
ι∈I
ι∈I
ι∈I
ι∈I
(3) (10 points)
Let A1 , A2 , . . . ⊂ Rn be a sequence of subsets.
(a) Show that
∞ [
∞
\
lim sup Ai :=
Ai = {x : x ∈ Ai for infinitely many k}.
i→∞
j=1
i=j
(b) What is the analogous description of
∞ \
∞
[
lim inf Ai :=
Ai
?
i→∞
j=1
i=j
(c) Prove that lim inf Ai ⊂ lim sup Ai .
(d) Prove that (lim sup Ai )c = lim inf Aci . (Hint: apply De Morgan’s Law and
(a),(b)).
(e) Let Bi ⊂ Rn be another sequence. Prove that lim sup(Ai ∪Bi ) = lim sup Ai ∪
lim sup Bi .
(f) Prove that lim inf(Ai ∩ Bi ) = lim inf Ai ∩ lim inf Bi .
(Hint: apply (d) and (e)).
(g) We say that the sequence Ai is decreasing if A1 ⊃ A2 ⊃ . . .. Prove that if
Ai is decreasing, then
lim sup Ai = lim inf Ai =
∞
\
Ai .
i=1
(h) Prove the analogous result for an increasing sequence of sets.
(4) (10 points)
Recall that a set A is called countable if there is a bijective (i.e.
one-to-one) map f : N → A.
(a) Prove that if A is countable and B ⊂ A, then B is countable.
(b) Prove that if B = A1 ∪ A2 and A1 , A2 are countable, then B is countable.
Conclude that Z (the set of integers) is countable.
3
(c) Prove that N2 is countable. Conclude from this that Nn is countable for all
n and that if A1 , . . . , An are countable, then so is A1 × . . . × An . (Hint: use
the fact that (x, y) 7→ 2x 3y is injective.)
(d) Prove that if B = A1 ∪ A2 ∪ A3 ∪ . . . and all Ai are countable, then B is
countable. Conclude from this that Q is countable (Hint: enumerate the
elements of B by two natural numbers).
(e) Let S ⊂ R be the set of all real numbers 0 ≤ x < 1 whose decimal expansion
only consists of zeros and ones, i.e. x = 0.p1 p2 p3 . . . where pi ∈ {0, 1}. Show
that S is not countable and hence [0, 1] and R are not countable.
Hint: Assume that there is a bijective f : N → S. Consider the number
y = 0.q1 q2 q3 . . . where
(
0 if f (i) has a 1 at its i-th decimal place
qi =
1 if f (i) has a 0 at its i-th decimal place
(5) (10 points)
T
(a) Give an example of open subsets A1 , A2 , . . . ⊂ Rn such that ∞
i=1 Ai is not
open.
S
(b) Give an example of closed subsets A1 , A2 , . . . ⊂ Rn such ∞
i=1 Ai is not
closed.
(c) Show that A ⊂ Rn is open if and only if for every sequence xi ∈ Rn with
limi→∞ xi = x and x ∈ A, there is a k such that xi ∈ A for all i > k.
(d) Show that A ⊂ Rn is closed if and only if for every sequence xi ∈ A with
limi→∞ xi = x, we have x ∈ A.
(e) Prove that for any A ⊂ Rn
[
\
A◦ =
B
and
A=
B
B⊂A,B open
A⊂B⊂Rn ,B closed
(Remark: For this and the following subproblem you can use either the
definition for the closure from the book or from class. Please indicate your
choice!)
c
(f) Show that for any A ⊂ Rn , we have A = (Ac )◦ and Ac = (A◦ )c .
(g) Show that if A ⊂ Rn is open and closed, then A = ∅ or Rn .
(Hint: Show this first for n = 1 and then use the fact that f −1 (A) is open
and closed for any continuous f : R → Rn .)
(6) (10 points)
Let A ⊂ Rn . In this problem we will mainly establish the equivalence of the following three conditions:
(i) A is closed and bounded (i.e. compact).
S
(ii) For any collection of open sets Uι ⊂ Rn , ι ∈ I with A ⊂ ι∈I Uι there is a
finite subcollection Uι1 , . . . , Uιk such that A ⊂ Uι1 ∪ . . . ∪ Uιk .
(iii) Every sequence x1 , x2 , . . . ∈ A has a subsequence xj1 , xj2 , . . . (with j1 < j2 <
. . .) such that limi→∞ xji exists and lies in A.
(a) Show first that each of conditions (ii) and (iii) imply (i).
(b) Show that condition (i) implies (iii)
4
(Hint: Subdivide A into smaller sets of diameter 2−k > 0, choose a subsequence which eventually stays in one of these subsets for all indices greater
than k, let k go to infinity)
S
(c) Assume that A satisfies condition (i) and consider a covering A ⊂ ι∈I Uι by
open subsets Uι ⊂ Rn . Show that the covering has a Lebesgue Number,
i.e. a number ε > 0 such that for every x ∈ A, the ball B(x, ε) is fully
contained in one of the Uι .
(Hint: Assume that this was wrong, then we can find a sequence xk ∈ A
such that B(xk , k1 ) is not fully contained in any of the Uι . Show that this
contradicts condition (iii)).
(d) Show that condition (i) implies (ii).
(Hint: Use (c) and cover A by finitely many ε-balls).
(7) (10 points)
Let O ⊂ P(Rn ) be the set of open subsets and F ⊂ P(Rn )
the set of special polygons (i.e. finite unions of special rectangles of the form
[a1 , b1 ) × . . . × [an , bn )).
(a) Show that for every P0 ∈ F.
λ2 (P0◦ ) := sup{λ1 (P ) : P ⊂ P0 , P ∈ F} = λ1 (P0 ).
(b) Let a < b and f : [a, b] → [0, ∞) be a continuous function. Let D =
{(x, y) ∈ R2 : a < x < b and 0 < y < f (x)} be the open region below the
graph of f . Show that
Z b
λ2 (D) =
f (x)dx,
a
where the latter is the Riemann integral.
(c) Show that in R2 we have: λ2 (Br (0)) = πr2 .
Maximum total points: 80