Measure, Integration & Real Analysis, 2021a

Section 5C Lebesgue Integration on R n 145
8 Show that the open unit ball in R n is an open subset of R n .
9 Suppose G 1 is a nonempty subset of R m and G 2 is a nonempty subset of R n .
Prove that G 1 × G 2 is an open subset of R m × R n if and only if G 1 is an open
subset of R m and G 2 is an open subset of R n .
[One direction of this result was already proved (see 5.36); both directions are
stated here to make the result look prettier and to be comparable to the next
exercise, where neither direction has been proved.]
10 Suppose F 1 is a nonempty subset of R m and F 2 is a nonempty subset of R n .
Prove that F 1 × F 2 is a closed subset of R m × R n if and only if F 1 is a closed
subset of R m and F 2 is a closed subset of R n .
11 Suppose E is a subset of R m × R n and
A = {x ∈ R m : (x, y) ∈ E for some y ∈ R n }.
(a) Prove that if E is an open subset of R m × R n , then A is an open subset
of R m .
(b) Prove or give a counterexample: If E is a closed subset of R m × R n , then
A is a closed subset of R m .
12 (a) Prove that lim n→∞ λ n (B n )=0.
(b) Find the value of n that maximizes λ n (B n ).
13 For readers familiar with the gamma function Γ: Prove that
for every positive integer n.
14 Define f : R 2 → R by
λ n (B n )=
πn/2
Γ( n 2 + 1)
⎧
⎨ xy(x 2 − y 2 )
f (x, y) = x
⎩
2 + y 2 if (x, y) ̸= (0, 0),
0 if (x, y) =(0, 0).
(a) Prove that D 1 (D 2 f ) and D 2 (D 1 f ) exist everywhere on R 2 .
(b) Show that ( D 1 (D 2 f ) ) (0, 0) ̸= ( D 2 (D 1 f ) ) (0, 0).
(c) Explain why (b) does not violate 5.48.
Measure, Integration & Real Analysis, by Sheldon Axler