The Vitali covering theorem is a profound result about coverings of a set in ℝ^{p} with open
balls. The balls can be defined in terms of any norm for ℝ^{p}. For example, the norm could
be
||x|| ≡ max{|xk| : k = 1,⋅⋅⋅,p}
or the usual norm
∘ ∑-------
|x| = |xk|2
k
or any other. The proof given here is from Basic Analysis [20]. Before beginning the proof,
here is a useful lemma.
Lemma 9.3.1In a normed linear space,
------
B (x,r) = {y : ∥y − x∥ ≤ r}
Proof: It is clear that B
(x,r)
⊆
{y : ∥y− x ∥ ≤ r}
because if y ∈B
(x,r)
, then there
exists a sequence of points of B
(x,r)
,
{xn}
such that
∥xn − y ∥
→ 0,
∥xn∥
< r. However, this
requires that
∥xn∥
→
∥y∥
and so
∥y∥
≤ r. Now let y be in the right side. It suffices to
consider
∥y − x ∥
= 1. Then you could consider for t ∈
(0,1)
, x + t
(y− x )
= z
(t)
.
Then
∥z(t)− x ∥ = t∥y − x ∥ = tr < r
and so z
(t)
∈ B
(x,r)
. But also,
∥z (t)− y∥ = (1− t)∥y− x∥ = (1 − t)r
so lim_{t→0}
∥z(t) − y∥
= 0 showing that y ∈B
(x,r)
. ■
Thus the usual way we think about the closure of a ball is completely correct in a normed
linear space. Recall that this lemma is not always true in the context of a metric space. Recall
the discrete metric for example in which the distance between different points is 1 and
distance between a point and itself is 0. In what follows we will use the result of this lemma
without comment.
Lemma 9.3.2 Let ℱ be a countable collection of open balls satisfying
∞ > M ≡ sup{r : B (p,r) ∈ ℱ} > 0
and let k ∈
(0,∞ )
. Then there existsG⊆ℱsuch that
If B (p,r) ∈ Gthen r > k, (9.5)
(9.5)
--- ---
If B1,B2 ∈ G thenB1 ∩B2 = ∅, (9.6)
(9.6)
G is maximal with respect to 9.5 and 9.6. (9.7)
(9.7)
By this is meant that if ℋ is a collection of balls satisfying 9.5and 9.6, then ℋ cannotproperly contain G.
Proof: If no ball of ℱ has radius larger than k, let G = ∅. Assume therefore, that some
balls have radius larger than k. Let ℱ≡
{Bi}
_{i=1}^{∞}. Now let B_{n1} be the first ball in the list
which has radius greater than k. If every ball having radius larger than k has closure which
intersects B_{n1}, then stop. The maximal set is
{Bn1}
. Otherwise, let B_{n2} be the next ball
having radius larger than k for which B_{n2}∩B_{n1} = ∅. Continue this way obtaining
{Bni}
_{i=1}^{∞}, a finite or infinite sequence of balls having radius larger than k whose closures
are disjoint. Then let G≡
{Bni}
. To see G is maximal with respect to 9.5 and 9.6,
suppose B ∈ℱ, B has radius larger than k, and G∪
{B }
satisfies 9.5 and 9.6.
Then at some point in the process, B would have been chosen because it would be
the ball of radius larger than k which has the smallest index at some point in the
construction. Therefore, B ∈G and this shows G is maximal with respect to 9.5 and 9.6.
■
Proposition 9.3.3Let ℱ be a collection of open balls, and let
A ≡ ∪ {B : B ∈ ℱ }.
Suppose
∞ > M ≡ sup {r : B (p,r) ∈ ℱ} > 0.
Then there exists G⊆ℱ such that G consists of balls whose closures are disjointand
A ⊆ ∪ {B^ : B ∈ G }
where for B = B
(x,r)
a ball,
B^
denotes the open ball B
(x,5r)
.
Proof: First of all, it follows from Theorem 2.4.5 on Page 63 that there is a countable
subset of ℱ which also covers A. Thus it can be assumed that ℱ is countable.
By Lemma 9.3.2, there exists G_{1}⊆ℱ which satisfies 9.5, 9.6, and 9.7 with k =
2M3--
.
Suppose G_{1},
⋅⋅⋅
,G_{m−1} have been chosen for m ≥ 2. Let G_{i} denote the collection of closures
of the balls of G_{i}. Then let ℱ_{m} be those balls of ℱ, such that if B is one of these balls, B has
empty intersection with every closed ball of G_{i} for each i ≤ m− 1. Then using Lemma 9.3.2,
let G_{m} be a maximal collection of balls from ℱ_{m} with the property that each ball has
radius larger than
( )
23
^{m}M and their closures are disjoint. Let G≡∪_{k=1}^{∞}G_{k}. Thus
the closures of balls in G are disjoint. Let x ∈ B
(p,r)
∈ℱ∖G. Choose m such
that
( )m ( )m −1
2 M < r ≤ 2 M
3 3
Then B
(p,r)
must have nonempty intersection with the closure of some ball from
G_{1}∪
⋅⋅⋅
∪G_{m} because if it didn’t, then G_{m} would fail to be maximal. Denote by B
(p0,r0)
a ball in G_{1}∪
⋅⋅⋅
∪G_{m} whose closure has nonempty intersection with B
(p,r)
.
Thus
( )
2 m
r0,r > 3 M.
Consider the picture, in which w ∈B
(p0,r0)
∩B
(p,r)
.
PICT
Then for x ∈B
(p,r)
,
--≤r0--
|∥x − p ∥ ≤ ∥x − p∥+ ∥p − w∥ + ◜∥w ◞◟− p ◝∥
0 0
<32r0 <r
◜(--)◞m◟−1-◝ ( )◜(--◞)◟m0-◝
≤ r +r + r ≤ 2 2 M + r ≤ 2 3 2 M + r ≤ 4r
0 3 0 2 3 0 0
Thus B
(p,r)
is contained in B
(p0,4r0)
. It follows that the closures of the balls of G are
disjoint and the set
{ ˆ }
B : B ∈ G
covers A. ■
Here is the concept of a Vitali covering.
Definition 9.3.4Let S be a set and let C be a covering of S meaning that everypoint of S is contained in a set of C. This covering is said to be a Vitali coveringif foreach ε > 0 and x ∈ S, there exists a set B ∈C containing x, the diameter of B is lessthan ε, and there exists an upper bound to the set of diameters of sets of C.
Recall the outer measure determined by a measure explained in Theorem 9.1.4. When the
measure is m_{p} that is p dimensional Lebesgue measure, denote this outer measure as m_{p}.
Thus
¯mp (S) ≡ inf{mp (F ) : F ⊇ S,F measurable}
Recall that m_{p} = m_{p} on the σ algebra of Lebesgue measurable sets. Also, if E is any set,
there exists F ⊇ E such that F is measurable and m_{p}
(F )
= m_{p}
(E)
. Then the following is
also called the Vitali covering theorem.
Theorem 9.3.5Let E ⊆ ℝ^{p}be a bounded set and let ℱ be a collection of openballs, of boundedradii such that ℱ covers E in the sense of Vitali. Then there exists acountable collection of balls from ℱ whose closures are disjoint, denoted by {B_{j}}_{j=1}^{∞},such thatm_{p}(E ∖∪_{j=1}^{∞}B_{j}) = m_{p}(E ∖∪_{j=1}^{∞}B_{j}) = 0.
Proof: From the definition of Lebesgue measure,
mp (B(x,αr)) = mp (B (0,αr))
= αpm (0,r) = αpm (B (x,r)),
p p
This is especially clear if the norm is
∥⋅∥
_{∞} because in this case, the balls are just
p dimensional cubes centered at x. It is also true for any other norm, which will
be made clear later after the change of variables formula has been presented. Let
S
= 0, there is nothing to prove, so assume the outer measure of this set is
positive. Let F ⊇ E such that F is measurable and m_{p}
(F)
= m_{p}
(E )
. By outer regularity of
Lebesgue measure, there exists U, an open set which satisfies
mp (F) > (1 − 10− p)mp (U ),U ⊇ F.
PICT
Each point of F is contained in balls of ℱ of arbitrarily small radii and so there exists a
covering of F with balls of ℱ whose closures are contained in U. Therefore, by
Proposition 9.3.3, there exist balls,
{Bi}
_{i=1}^{∞}⊆ℱ such that their closures are disjoint
and
---
F ⊆ ∪∞j=1^Bj,Bj ⊆ U.
Therefore,
m (F ∖∪ ∞ B--) ≤ m (U )− m (∪∞ B-)
p j=1 j p p j=1 j ∞
( −p)−1 ∑ (--)
< 1− 10 mp (F)− j=1mp Bj
∞ ( )
= (1− 10−p)−1m (F)− 5−p ∑ m ^B
p j=1 p j
( −p)−1 −p
≤ 1− 10 mp (F)− 5 mp (F)
= mp (F )θp
Now consider F ∖∪_{j=1}^{m1}B_{j} and apply the same reasoning to it that was done to F. Thus
there exists m_{2}> m_{1} such that
m (F ∖∪m2 B--) < θ m (F ∖ ∪m1 B-) < θ2m (F)
p j=1 j p p j=1 j p p
Continuing this way, there exists an increasing subsequence m_{k} such that
( --)
mp F ∖ ∪mjk=1Bj < θkpmp (F)
and since θ_{p}< 1, and m_{p}
(F )
< ∞, this implies m_{p}
(F ∖∪ ∞ B--)
j=1 j
= 0. Now
0 ≤ ¯mp (E ∖ ∪∞j=1Bj) ≤ ¯mp(E ∖ ∪∞j=1Bj)
( ∞ ) ( ∞ --)
≤ mp F ∖ ∪j=1Bj = mp F ∖ ∪j=1Bj = 0 ■
You don’t need to assume that E is bounded in order to draw the above conclusion.
Corollary 9.3.6Let E ⊆ ℝ^{p}be a setand let ℱ be a collection of balls which comefrom some norm, open or not, but having bounded radii such that ℱ covers E in thesense of Vitali. Thenthere exists a countable collection of balls from ℱ having disjointclosures, denoted by {B_{j}}_{j=1}^{∞}, such thatm_{p}(E ∖∪_{j=1}^{∞}B_{j}) = 0.
Proof: Consider A_{n} = B
(0,n)
∖B
(0,n − 1)
,n = 1,2,
⋅⋅⋅
. Let E_{n} = E ∩ A_{n}. Then
∪_{n=1}^{∞}E_{n}∪N = E where N is a set of measure zero. From Theorem 9.3.5, there exist balls of
ℱ, having disjoint closures denoted by
{Bni }
_{i=1}^{∞}, such that m_{p}
(En ∖∪ ∞i=1Bni )
= 0 and each
B_{i}^{n}⊆ A_{n}. Then
{Bni ,(i,n ) ∈ ℕ× ℕ }
is a suitable collection of balls having disjoint closures.
In fact,
∑
m¯p (E ∖∪n ∪∞i=1 Bni ) = ¯mp (∪n(En ∖∪∞i=1Bni )) ≤ ¯mp (En ∖∪ ∞i=1Bni ) = 0.■
n