There is another covering theorem which may also be referred to as the Besicovitch covering theorem. As
before, the balls can be taken with respect to any norm on ℝ^{n}. At first, the balls will be closed but this
assumption will be removed.
Definition 21.4.1A collection of balls, ℱ covers a set, E in the sense of Vitali if whenever x ∈ Eand ε > 0, there exists a ball B ∈ℱ whose center is x having diameter less than ε.
I will give a proof of the following theorem.
Theorem 21.4.2Let μ be a Radon measure on ℝ^{n}and let E be a set with μ
(E )
< ∞. Where μis theouter measure determined by μ. Suppose ℱ is a collection of closed balls which cover E in the sense ofVitali. Then there exists a sequence of disjoint balls,
{Bi}
⊆ℱ such that
( )
μ-E ∖∪ ∞j=1Bj = 0.
Proof:Let N_{n} be the constant of the Besicovitch covering theorem. Choose r > 0 such
that
( )
(1 − r)−1 1− ---1--- ≡ λ < 1.
2Nn + 2
If μ
(E )
= 0, there is nothing to prove so assume μ
(E)
> 0. Let U_{1} be an open set containing E with
(1 − r)
μ
(U1)
< μ
(E )
and 2μ
(E )
> μ
(U1)
, and let ℱ_{1} be those sets of ℱ which are contained in U_{1} whose
centers are in E. Thus ℱ_{1} is also a Vitali cover of E. Now by the Besicovitch covering theorem proved
earlier, there exist balls B, of ℱ_{1} such that
Nn
E ⊆ ∪i=1 {B : B ∈ Gi}
where G_{i} consists of a collection of disjoint balls of ℱ_{1}. Therefore,
Nn
μ(E) ≤ ∑ ∑ μ(B)
i=1 B∈Gi
and so, for some i ≤ N_{n},
∑
(Nn + 1) μ(B ) > μ(E ).
B∈Gi
It follows there exists a finite set of balls of G_{i},
m1
≥ μ (U )− ∑ μ(B ) = μ(U )− μ(∪m1 B )
1 i=1 i 1 j=1 j
= μ (U1 ∖ ∪m1 Bj) ≥ μ-(E ∖∪m1 Bj).
j=1 j=1
Since the balls are closed, you can consider the sets of ℱ which have empty intersection with ∪_{j=1}^{m1}B_{j}
and this new collection of sets will be a Vitali cover of E ∖∪_{j=1}^{m1}B_{j}. Letting this collection of balls play
the role of ℱ in the above argument and letting E ∖∪_{j=1}^{m1}B_{j} play the role of E, repeat the above
argument and obtain disjoint sets of ℱ,
Continuing in this way, yields a sequence of disjoint balls
{B }
i
contained in ℱ and
μ-(E ∖∪ ∞j=1Bj ) ≤ μ-(E ∖∪mkj=1Bj ) < λkμ-(E )
for all k. Therefore, μ
( ∞ )
E ∖∪j=1Bj
= 0 and this proves the Theorem. ■
It is not necessary to assume μ
(E )
< ∞.
Corollary 21.4.3Let μ be a Radon measure on ℝ^{n}. Letting μbe the outer measure determined by μ,suppose ℱ is a collection of closed balls which cover E in the sense of Vitali. Then there exists a sequenceof disjoint balls,
{Bi }
⊆ℱ such that
( )
μ-E ∖∪ ∞j=1Bj = 0.
Proof: Since μ is a Radon measure it is finite on compact sets. Therefore, there are at most countably
many numbers,
{bi}
_{i=1}^{∞} such that μ
(∂B (0,bi))
> 0. It follows there exists an increasing
sequence of positive numbers,
{ri}
_{i=1}^{∞} such that lim_{i→∞}r_{i} = ∞ and μ
(∂B (0,ri))
= 0. Now let
D1 ≡ {x : ||x|| < r1},D2 ≡ {x : r1 < ||x|| < r2},
⋅⋅⋅,D ≡ {x : r < ||x|| < r },⋅⋅⋅.
m m−1 m
Let ℱ_{m} denote those closed balls of ℱ which are contained in D_{m}. Then letting E_{m} denote E ∩D_{m}, ℱ_{m} is
a Vitali cover of E_{m},μ
(Em )
< ∞, and so by Theorem 21.4.2, there exists a countable sequence of balls
from ℱ_{m}
{ m }
Bj
_{j=1}^{∞}, such that μ
( ∞ m)
Em ∖∪j=1B j
= 0. Then consider the countable collection of balls,
{ m }
Bj
_{j,m=1}^{∞}.
μ(E ∖∪ ∞ ∪∞ Bm ) ≤ μ-(∪ ∞ ∂B (0,r )) +
∞ m=1 j=1 j j=1 i
∑ -( ∞ m)
+m=1 μ Em ∖∪j=1Bj = 0 ■
You don’t need to assume the balls are closed. In fact, the balls can be open, closed or anything in
between and the same conclusion can be drawn.
Corollary 21.4.4Let μ be a Radonmeasure on ℝ^{n}. Letting μbe the outer measure determined by μ,suppose ℱ is a collection of balls which cover E in the sense of Vitali, open closed or neither. Then thereexists a sequence of disjoint balls,
{Bi}
⊆ℱ such that
-( ∞ )
μ E ∖∪ j=1Bj = 0.
Proof: Let x ∈ E. Thus x is the center of arbitrarily small balls from ℱ. Since μ is a Radon measure,
at most countably many radii, r of these balls can have the property that μ
(∂B (0,r))
= 0. Let
ℱ^{′} denote the closures of the balls of ℱ, B
(x,r)
with the property that μ
(∂B (x,r))
= 0.
Since for each x ∈ E there are only countably many exceptions, ℱ^{′} is still a Vitali cover of E.
Therefore, by Corollary 21.4.3 there is a disjoint sequence of these balls of ℱ^{′},
{--}
Bi
_{i=1}^{∞} for
which
μ(E ∖ ∪∞ B-) = 0
j=1 j
However, since their boundaries have μ measure zero, it follows