Download 4. Homework Assignment #4 Math 4/515 Problem 4.1. If x > 0 is

4. Homework Assignment #4 Math 4/515
Problem 4.1. If x > 0 is rational, we may express x = pq for p, q ∈ N. We say this is in lowest terms if q is minimal.
Adopt this notation to represent any non-negative rational number uniquely. Let
1
if x is rational
q
if x ≥ 0 .
α(x) :=
0 if x is irrational or if x = 0
Let A = [0, 1] × [0, 1]. Let f (x, y) = α(x) + α(y) be a bounded map from A to [0, 2].
R
(1) Prove or disprove the assertion “f is integrable and A f = 0.” ( You may not use the Theorem that “f is
integrable if and only if it is continuous almost everywhere”).
(2) Determine the set of points in A where f is discontinuous.
Problem 4.2. Let {rn }n∈N be an enumeration of the rationals in the open interval (0, 1). Choose δn > 0 so that
In := (rn − δn , rn + δn ) ⊂ (0, 1) and δn < 71−n . Let O := ∪∞
n=1 In . This is an open subset of (0, 1).
(1) Show that the boundary of O is the closed set C := [0, 1] − O.
(2) Show that C does not have content 0. Hint. Suppose C had content 0. Cover C by a finite number
1
. The rectangles {R1 , . . . , Rk , I1 , . . . } forms a
of rectangles R1 ,. . . ,Rk so Vol(R1 ) + · · · + Vol(Rk ) < 71
countable cover of [0, 1]. Extract a finite sub cover. Thus for some n, we have the inclusion [0, 1] ⊂
R1 ∪ · · · ∪ Rk ∪ I1 ∪ · · · ∪ In . Try to show that this can not be true by integrating the inequality:
χ[0,1] ≤ χR1 + · · · + χRk + χI1 + · · · + χIn .
(3) Prove or disprove the following assertion: “Every open subset of (0, 1) is Jordan measurable.”
Problem 4.3. Let R = [0, 1] × [0, 1] ⊂ R2 .
(1) Construct a countable set S ⊂ R so that S̄ = R and so that any horizontal or vertical line contains at
most one point of S. (Hint: Let {pn }n∈N = {2, 3, 5, 7, 11, . . . } be the set of all prime numbers enumerated
in increasing order. Let {(xn , yn )}n∈N be an enumeration of the points in R which have both coordinates
√
rational. Let x̃n := xn ± 1/( pn · 69n ) where one chooses a sign to ensure that 0 < x̃n < 1; similarly let
√
ỹn = yn ± 1/( pn · 69n ) where a (perhaps different) sign is chosen to ensure that 0 < ỹn < 1. Show that
S = {(x̃n , ỹn )}n∈N has the desired property.)
(2) Let F = χS be the characteristic function of S. Show that F is not integrable
on R.
R
(3) Let ξx (y) := F (x, y). Show that ξx is integrable in y for any x and [0,1] ξx (y)dy = 0. Show the iterated
R1 R1
integral x=0 y=0 F (x, y)dydx = 0.
R
(4) Let ηy (x) := F (x, y). Show that ηy is integrable in x for any y and ηy (x)dx = 0. Show the integrated
R1 R1
integral y=0 x=0 F (x, y)dxdy = 0.
Problem 4.4. Let R = [0, 1] × [0, 1] ⊂ R2 . Define

69 if



−69 if
F (x, y) =
−69 if



69 if
x
x
x
x
is
is
is
is
rational and
rational and
irrational and
irrational and
(1) Prove or disprove the following assertion: “The iterated integral
y ∈ [0, 21 ]
y ∈ ( 12 , 1]
y ∈ [0, 12 ]
y ∈ ( 12 , 1]
R1 R1
Rx=0
Ry=0
1
1
(2) Prove or disprove the following assertion: “The iterated integral y=0
(3) Prove or disprove the following assertion: “F is integrable on R ”.
Problem 4.5. Correct any misprints as intelligently as possible.
1
x=0




.



F (x, y)dydx exists.”
F (x, y)dxdy exists.”