Measure, Integration & Real Analysis, 2021a

276 Chapter 9 Real and Complex Measures
If E ∈Sand μ(E) =0, then χ E
is the 0 element of L p (μ), which implies that
ϕ(χ E
)=0, which means that ν(E) =0. Hence ν ≪ μ. By the Radon–Nikodym
Theorem (9.36), there exists h ∈L 1 (μ) such that dν = h dμ. Hence
∫ ∫
ϕ(χ E
)=ν(E) = h dμ = χ E
h dμ
for every E ∈S. The equation above, along with the linearity of ϕ, implies that
∫
9.43 ϕ( f )= fhdμ for every simple S-measurable function f : X → F.
Because every bounded S-measurable function is the uniform limit on X of a
sequence of simple S-measurable functions (see 2.89), we can conclude from 9.43
that
∫
9.44 ϕ( f )= fhdμ for every f ∈ L ∞ (μ).
For k ∈ Z + , let
and define f k ∈ L p (μ) by
9.45 f k (x) =
E k = {x ∈ X :0< |h(x)| ≤k}
E
{
h(x) |h(x)| p′ −2
if x ∈ E k ,
0 otherwise.
Now
∫
(∫
|h| p′ χ
Ek
dμ = ϕ( f k ) ≤‖ϕ‖‖f k ‖ p = ‖ϕ‖
|h| p′ χ
Ek
dμ) 1/p,
where the first equality follows from 9.44 and 9.45, and the last equality follows from
9.45 [which implies that | f k (x)| p = |h(x)| p′ χ
Ek
(x) for x ∈ ( X]. After dividing by
∫ ) 1/p,
|h|
p ′ χ
Ek
dμ the inequality between the first and last terms above becomes
‖hχ
Ek
‖ p ′ ≤‖ϕ‖.
Taking the limit as k → ∞ shows, via the Monotone Convergence Theorem (3.11),
that
‖h‖ p ′ ≤‖ϕ‖.
Thus h ∈ L p′ (μ). Because each f ∈ L p (μ) can be approximated in the L p (μ) norm
by functions in L ∞ (μ), 9.44 now shows that ϕ = ϕ h , completing the proof in the
case where μ is a finite (positive) measure.
Now relax the assumption that μ is a finite (positive) measure to the hypothesis
that μ is a (positive) measure. For E ∈S, let S E = {A ∈S: A ⊂ E} and let μ E
be the (positive) measure on (E, S E ) defined by μ E (A) =μ(A) for A ∈S E .We
can identify L p (μ E ) with the subspace of functions in L p (μ) that vanish (almost
everywhere) outside E. With this identification, let ϕ E = ϕ| L p (μ E ) . Then ϕ E is a
bounded linear functional on L p (μ E ) and ‖ϕ E ‖≤‖ϕ‖.
Measure, Integration & Real Analysis, by Sheldon Axler