29.4 Vitali Coverings
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 ℝn. At
first, the balls will be closed but this assumption will be removed.
Definition 29.4.1 A collection of balls, ℱ covers a set, E in the sense of Vitali
if whenever x ∈ E and 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 29.4.2 Let μ be a Radon measure on ℝn and let E be a set with μ
Where μ is 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 sequence of disjoint balls,
⊆ℱ such that
Proof: Let Nn be the constant of the Besicovitch covering theorem. Choose r > 0
there is nothing to prove so assume μ
0. Let U1
be an open set
and let ℱ1
be those sets
which are contained in U1
whose centers are in E.
is also a Vitali cover of
Now by the Besicovitch covering theorem proved earlier, there exist balls, B,
where Gi consists of a collection of disjoint balls of ℱ1. Therefore,
and so, for some i ≤ Nn,
It follows there exists a finite set of balls of Gi,
Also U1 was chosen such that
Since the balls are closed, you can consider the sets of ℱ which have empty intersection
with ∪j=1m1Bj and this new collection of sets will be a Vitali cover of E ∖∪j=1m1Bj.
Letting this collection of balls play the role of ℱ in the above argument and letting
E ∖∪j=1m1Bj play the role of E, repeat the above argument and obtain disjoint sets of
Continuing in this way, yields a sequence of disjoint balls
for all k. Therefore, μ
= 0 and this proves the Theorem.
It is not necessary to assume μ
Corollary 29.4.3 Let μ be a Radon measure on ℝn. Letting μ be 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 sequence of disjoint balls,
Proof: Since μ is a Radon measure it is finite on compact sets. Therefore, there are at
most countably many numbers,
such that μ
It follows there
exists an increasing sequence of positive numbers,
such that limi→∞ri
denote those closed balls of ℱ
which are contained in Dm.
Then letting Em
denote E ∩ Dm, ℱm
is a Vitali cover of Em,μ
and so by Theorem
, there exists a countable sequence of balls from ℱm
Then consider the countable collection of balls,
This proves the corollary.
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 29.4.4 Let μ be a Radon measure on ℝn. Letting μ be 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 there exists a sequence of disjoint balls,
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
denote the closures of the balls of ℱ
with the property that μ
Since for each x ∈ E
only countably many exceptions, ℱ′
is still a Vitali cover of E.
there is a disjoint sequence of these balls of ℱ′
However, since their boundaries have μ measure zero, it follows
This proves the corollary.