When dealing with probability distribution functions or some other Radon measure, it is necessary to have a better covering theorem than the Vitali covering theorem which works well for Lebesgue measure. However, for a Radon measure, if you enlarge the ball by making the radius larger, you don’t know what happens to the measure of the enlarged ball except that its measure does not get smaller. Thus the thing required is a covering theorem which does not depend on enlarging balls.
The first fundamental observation is found in the following lemma which holds for the context illustrated by the following picture. This picture is drawn such that the balls come from the usual Euclidean norm, but the norm could be any norm on ℝ^{n}.
The idea is to consider balls B_{i} which intersect a given ball B such that B contains no center of any B_{i} and no B_{i} contains the center of another B_{j}. There are two cases to consider, the case where the balls have large radii and the case where the balls have small radii.
Intersections with big balls
Lemma 29.1.1 Let the balls B_{a},B_{x},B_{y} be as shown having radii r,r_{x},r_{y} respectively. Suppose the centers of B_{x} and B_{y} are not both in any of the balls shown, and suppose r_{y} ≥ r_{x} ≥ αr where α is a number larger than 1. Also let P_{x} ≡ a+r
Proof: From the definition,




 (29.1.1) 
There are two cases. First suppose that

From the assumptions,
The other case is that

29.1.1 equals
Finally, in the case of the balls B_{i} having centers at x_{i}, then as above, let P_{xi} = a + r

How many points on the unit sphere can be pairwise this far apart? This set is compact and so there exists a
The above lemma has do do with balls which are relatively large intersecting a given ball. Next is a lemma which has to do with relatively small balls intersecting a given ball. Note that in the statement of this lemma, the radii are smaller than αr in contrast to the above lemma in which the radii of the balls are larger an αr. In the application of this lemma, we will have γ = 4∕3 and β = 1∕3. These constants will come from a construction, while α is just something larger than 1 which we will take here to equal 10.
Intersections with small but comparable balls
Lemma 29.1.2 Let B be a ball having radius r and suppose B has nonempty intersection with the balls B_{1},

Let B_{i}^{′} have the same center as B_{i} with radius equal to r_{i}^{′} = βr_{i} for some β < 1. If the B_{i}^{′} are disjoint, then there exists a constant M
Proof: Let the volume of a ball of radius r be given by α
This can be done for a single B_{i}^{′} by enlarging the radius of B to r + r_{i} + r_{i}^{′}.
Then to get all the B_{i}, you would just enlarge the radius of B to r + αr + βαr =

Therefore,

and so m ≤
From now on, let α = 10 and let β = 1∕3 and γ = 4∕3. Then

Thus m ≤ 60^{n}. ■
The next lemma gives a construction which yields balls which are comparable as described in the above lemma. r
A construction of a sequence of balls
Lemma 29.1.3 Let ℱ be a nonempty set of nonempty balls in ℝ^{n} with

and let A denote the set of centers of these balls. Suppose A is bounded. Define a sequence of balls from ℱ,
 (29.1.2) 
and if
 (29.1.3) 
then B_{m+1} ∈ℱ is chosen with center in A_{m} such that
 (29.1.4) 
Then letting B_{j} = B
 (29.1.5) 
 (29.1.6) 
 (29.1.7) 
Proof: Consider 29.1.5. First note the sets A_{m} form a decreasing sequence. Thus from the definition of B_{j}, for j < k,

Next consider 29.1.6. If x ∈ B

and this contradicts the construction because a_{j} is not covered by B
Finally consider the claim that A ⊆∪_{i=1}^{J}B_{i}. Pick B_{1} satisfying 29.1.2. If B_{1},
As explained above, in this sequence of balls from the above lemma, if j < k

Then there are two cases to consider,

In the first case, we use Lemma 29.1.1 to estimate the number of intersections of B_{k} with B_{j} for j < k. In the second case, we use Lemma 29.1.2 to estimate the number of intersections of B_{k} with B_{j} for j < k.
Now here is the Besicovitch covering theorem.
Theorem 29.1.4 There exists a constant N_{n}, depending only on n with the following property. If ℱ is any collection of nonempty balls in ℝ^{n} with

and if A is the set of centers of the balls in ℱ, then there exist subsets of ℱ, ℋ_{1},

Proof: To begin with, suppose A is bounded. Let L
Now let R_{1} = B

Thus, if

Note that the balls in G_{j}^{′} are disjoint. This is because those in G_{j}