Measure, Integration & Real Analysis, 2021a

200 Chapter 7 L p Spaces
6 Suppose (X, S, μ) is a measure space, f ∈L 1 (μ), and h ∈L ∞ (μ). Prove that
‖ fh‖ 1 = ‖ f ‖ 1 ‖h‖ ∞ if and only if
|h(x)| = ‖h‖ ∞
for almost every x ∈ X such that f (x) ̸= 0.
7 Suppose (X, S, μ) is a measure space and f , h : X → F are S-measurable.
Prove that
‖ fh‖ r ≤‖f ‖ p ‖h‖ q
for all positive numbers p, q, r such that 1 p + 1 q = 1 r .
8 Suppose (X, S, μ) is a measure space and n ∈ Z + . Prove that
‖ f 1 f 2 ··· f n ‖ 1 ≤‖f 1 ‖ p1 ‖ f 2 ‖ p2 ··· ‖f n ‖ pn
1
for all positive numbers p 1 ,...,p n such that p1
+
p 1 2
+ ···+
p 1 n
S-measurable functions f 1 , f 2 ,..., f n : X → F.
= 1 and all
9 Show that the formula in 7.12 holds for p = ∞ if μ is a σ-finite measure.
10 Suppose 0 < p < q ≤ ∞.
(a) Prove that l p ⊂ l q .
(b) Prove that ‖(a 1 , a 2 ,...)‖ p ≥‖(a 1 , a 2 ,...)‖ q for every sequence a 1 , a 2 ,...
of elements of F.
11 Show that ⋂ p>1
l p ̸= l 1 .
12 Show that
⋂
p1
L p ([0, 1]) ̸= L 1 ([0, 1]).
14 Suppose p, q ∈ (0, ∞], with p ̸= q. Prove that neither of the sets L p (R) and
L q (R) is a subset of the other.
15 Show that there exists f ∈L 2 (R) such that f /∈ L p (R) for all p ∈ (0, ∞] \{2}.
16 Suppose (X, S, μ) is a finite measure space. Prove that
lim
p→∞ ‖ f ‖ p = ‖ f ‖ ∞
for every S-measurable function f : X → F.
17 Suppose μ is a measure, 0 < p ≤ ∞, and f ∈L p (μ). Prove that for every
ε > 0, there exists a simple function g ∈L p (μ) such that ‖ f − g‖ p < ε.
[This exercise extends 3.44.]
Measure, Integration & Real Analysis, by Sheldon Axler