Assignment 7 of Math 5302
1. Let A ⊂ R
n be a compact set. Show that A is bounded.
2. Let G be a nonempty subset of R
n
. If G is open and P is a special polygon with P ⊂ G, prove
there exists a special polygon P
0
such that P ⊂ P
0 ⊂ G and λ(P) < λ(P
0
). (Hint: consider G ∼ P.)
3. Use the definition of Lebesgue measure, λ(G), of an open set G ⊂ R
n
to prove the following
statements:
(a) If G is a bounded open set, then λ(G) < ∞.
(b) Let
G = {(x, y) ∈ R
2
: 0 < x < 1 and 0 < y < x2
}.
Then λ(G) = 1
3
. (Hint: relate λ(G) to lower and upper Darboux sums of the function f(x) = x
2 on
[0, 1]. However, you cannot use the methods of calculus to the extent that λ(G) = R 1
0
x
2dx =
1
3
. You
must use the actual definition of λ(G).)
4. Prove that every nonempty open subset of R can be expressed as a countable disjoint union of open
intervals:
G =
[
k
(ak, bk),
where the range on k can be finite or infinite. Furthermore, show that this expression is unique except
for the numbering of the component intervals. (Hint: for any x ∈ G, show that there exists a largest
open interval Ax such that x ∈ Ax and Ax ⊆ G. Also note that the set of rational numbers is countable
and dense in R.)
5. In the notation of Problem 4, prove that λ(G) = P
k
(bk − ak).
6. Let C be the Cantor set. Show that 1
4
∈ C and that 1
4
is not an end point of any of the intervals in
the Gk’s.
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