24.7 Differentiation Of Measures With Respect To Lebesgue Measure
Recall the Vitali covering theorem in Corollary 11.4.5 on Page 993.
Corollary 24.7.1Let E ⊆ ℝ^{n}and letℱ, be a collection of open balls of boundedradii such that ℱ covers E in the sense of Vitali. Then there exists a countablecollection of disjoint balls from ℱ, {B_{j}}_{j=1}^{∞}, such that m(E ∖∪_{j=1}^{∞}B_{j}) = 0.
Definition 24.7.2Let μ be a Radon measure defined on ℝ^{n}. Then
-dμ (x ) ≡ lim-μ(B-(x,r))
dm r→0m (B (x,r))
whenever this limit exists.
It turns out this limit exists for m a.e. x. To verify this here is another definition.
Definition 24.7.3Let f
(r)
be a function having values in
[− ∞, ∞ ]
. Then
lim sup f (r) ≡ lim (sup {f (t) : t ∈ [0,r]})
r→0+ r→0
lim rin→f0+f (r) ≡ rli→m0 (inf{f (t) : t ∈ [0,r]})
This is well defined because the function r → inf
{f (t) : t ∈ [0,r]}
is increasing andr → sup
{f (t) : t ∈ [0,r]}
is decreasing. Also note that lim_{r→0+}f
(r)
exists if and onlyif
limrs→u0p+ f (r) = lim ri→n0f+ f (r)
and if this happens
rli→m0+f (r) = lim r→in0f+ f (r) = lim sr→u0p+ f (r).
The claims made in the above definition follow immediately from the definition of
what is meant by a limit in
[− ∞, ∞ ]
and are left for the reader.
Theorem 24.7.4Let μ be a Borel measure on ℝ^{n}then
ddμm-
(x)
exists in
[− ∞, ∞ ]
m a.e.
Proof:Let p < q and let p,q be rational numbers. Define
{
N (M ) ≡ x ∈ ℝn such that lim sup μ(B-(x,r))-> q
pq r→0+ m (B (x,r))
μ(B (x,r)) }
> p > lim rin→f0+ m-(B-(x,r)) ∩ B (0,M ),
{
Npq ≡ x ∈ ℝn such that lim sup μ(B-(x,r))-> q
r→0+}m (B (x,r))
μ(B-(x,r))-
> p > lim rin→f0+ m (B (x,r)) ,
{ μ(B (x,r))
N ≡ x ∈ ℝn such that lim sup m-(B-(x,r)) >
} r→0+
lim inf μ-(B(x,r)) .
r→0+ m (B (x,r))
I will show m
(Npq(M ))
= 0. Use outer regularity to obtain an open set, V containing
N_{pq}
(M )
such that
--
m (Npq(M ))+ ε > m (V ).
From the definition of N_{pq}
(M )
, it follows that for each x ∈ N_{pq}
(M )
there exist
arbitrarily small r > 0 such that
μ(B (x,r))
---------- < p.
m (B (x,r))
Only consider those r which are small enough to be contained in B
(0,M )
so that the
collection of such balls has bounded radii. This is a Vitali cover of N_{pq}
(M )
and so by
Corollary 24.7.1 there exists a sequence of disjoint balls of this sort,
These are measures thanks to Theorem 18.2.3on Page 1834and μ^{+}− μ^{−} = μ. Thesemeasures have values in [0,∞). They are called the positive and negative parts of μrespectively.For μ a complex measure, defineReμ andImμ by
( )
Re μ(E ) ≡ 1 μ(E)+ μ-(E)
2 ( )
Im μ(E ) ≡ 1- μ (E )− μ(E-)
2i
ThenReμ andImμ are both real measures. Thus for μ a complex measure,
+ − ( + − )
μ = Re μ − Re μ + i Im μ − Im μ
= ν1 − ν1 + i(ν3 − ν4)
where each ν_{i}is a real measure having values in [0,∞).
Then there is an obvious corollary to Theorem 24.7.4.
Corollary 24.7.6Let μ be a complex Borel measure on ℝ^{n}. Then
ddμm-
(x)
existsa.e.
Proof: Letting ν_{i} be defined in Definition 24.7.5. By Theorem 24.7.4, for m a.e. x,
dνi
dm
(x)
exists. This proves the corollary because μ is just a finite sum of these
ν_{i}.
Theorem 18.1.2 on Page 1813, the Radon Nikodym theorem, implies that if you have
two finite measures, μ and λ, you can write λ as the sum of a measure absolutely
continuous with respect to μ and one which is singular to μ in a unique way. The next
topic is related to this. It has to do with the differentiation of a measure which is singular
with respect to Lebesgue measure.
Theorem 24.7.7Let μ be a Radon measure on ℝ^{n}and suppose there exists a μmeasurable set, N such that for all Borel sets, E, μ
(E )
= μ
(E ∩ N)
where m
(N )
= 0.Then
dμ-
dm (x) = 0 m a.e.
Proof: For k ∈ ℕ, let
{ }
Bk (M ) ≡ x ∈ N C : lim sup μ(B-(x,r))-> 1- ∩B (0,M ),
{ r→0+ m (B (x,r)) k }
C μ(B-(x,r))- 1-
Bk ≡ x ∈ N : lim rsu→p0+ m (B (x,r)) > k ,
{ μ(B (x,r)) }
B ≡ x ∈ N C : lim sup ----------> 0 .
r→0+ m (B (x,r))
Let ε > 0. Since μ is regular, there exists H, a compact set such that H ⊆ N ∩B