MATH 515 Supplementary Homework
Fall 2014
1. Let E ⊂ R be a set of measure zero. Show that the set E1 = {x2 : x ∈ E} also has
measure zero. (Suggestion: first show that E1 ∩ [0, n] has measure zero for any fixed n.)
2. Let E = [−1, 1] × [−1, 1] and
f (x, y) =
(x2
xy
+ y 2 )2
Show that the two iterated integrals of f over E are finite and equal, but f 6∈ L1 (E).
3. Let E = [0, 1] × [0, 1] and

2n

2
f (x, y) = −22n+1


0
2−n ≤ x < 21−n 2−n ≤ y < 21−n
2−n−1 ≤ x < 2−n 2−n ≤ y < 21−n
otherwise
Show that the two iterated integrals of f over E are finite but unequal.
4. Give a careful derivation of the formula
Z
√
2
e−x dx = π
R
by relating it to the double integral
ZZ
e−(x
2 +y 2 )
dx dy
R2
5. Find a set E and a function f on E such that
{p : f ∈ Lp (E)} = [2, 3)
(Suggestion: think about functions of the form
1
xα logβ x
for various choices of α and β.)
6. Prove the following generalization of Hölder’s inequality: if
k
X
1
1
=
p
r
j=1 j
1 ≤ pj ≤ ∞
then
||f1 f2 . . . fk ||r ≤ ||f1 ||p1 ||f2 ||p2 . . . ||fk ||pk
7. Use a density argument to show that if f ∈ L1 (a, b) then
Z
lim
n→∞
b
f (x) cos nx dx = 0
a
(This is a special case of the Riemann-Lebesgue lemma.)
8. If f is a measurable function on Rn define Rf , the essential range of f , to be the
set of y ∈ R such that {x ∈ E : |f (x) − y| < } has positive measure for all > 0. If
f ∈ L∞ (E) show that Rf is compact and
||f ||∞ = sup |y|
y∈Rf
Give an example showing the difference between the range of f and the essential range
of f .
9. Let 1 < p < ∞, and suppose φ is a continuous linear functional on Lp (R), such that
for every f ∈ Lp (R), φ(f (x)) = φ(f (x − 1)). Show that φ is the zero functional.