61.4. A MAJOR EXISTENCE AND CONVERGENCE THEOREM 1997

First I will describe a construction. Letting C ∈B (E) and r > 0,

Cr1 ≡ C∩B(a1,r) ,Cr

2 ≡ B(a2,r)∩C \Cr1, · · · ,

Crn ≡ B(an,r)∩C \

(∪n−1

k=1Crk).

Thus the sets, Crk for k = 1,2, · · · are disjoint Borel sets whose union is all of C. Of course

many may be empty.Cr(size)

n(index of the {ak} it is close to)

Now let C = E, the whole metric space. Also let {rk} be a decreasing sequence of positivenumbers which converges to 0. Let

Ak ≡ Er1k , k = 1,2, · · ·

Thus {Ak} is a sequence of Borel sets, Ak ⊆ B(ak,r1) , and the union of the Ak equals E.For (i1, · · · , im) ∈ Nm, suppose Ai1,··· ,im has been defined. Then for k ∈ N,

Ai1,··· ,imk ≡ (Ai1,··· ,im)rm+1k

Thus Ai1,··· ,imk ⊆ B(ak,rm+1), is a Borel set, and

∪∞k=1Ai1,··· ,imk = Ai1,··· ,im . (61.4.9)

Also note that Ai1,··· ,im could be empty. This is because Ai1,··· ,imk ⊆ B(ak,rm+1) but alsoAi1,··· ,im ⊆ B(aim ,rm) which might have empty intersection with B(ak,rm+1) . Applying61.4.9 repeatedly,

E = ∪i1 · · ·∪im Ai1,··· ,im

and also, the construction shows the Borel sets, Ai1,··· ,im are disjoint. Note that to getAi1,··· ,imk, you do to Ai1,··· ,im what was done for E but you consider smaller sized pieces.

Construction of intervals depending on the measure

Next I will construct intervals, Iνi1,··· ,in in [0,1) corresponding to these Ai1,··· ,in . In what

follows, ν = µn or µ . These intervals will depend on the measure chosen as indicated inthe notation.

Iν1 ≡ [0,ν (A1)), · · · , Iν

j ≡

[j−1

∑k=1

ν (Ak) ,j

∑k=1

ν (Ak)

)for j = 1,2, · · · . Note these are disjoint intervals whose union is [0,1). Also note

m(Iν

j)= ν (A j) .

The endpoints of these intervals as well as their lengths depend on the measures of the setsAk. Now supposing Iν

i1,··· ,im = [α,β ) where β −α = ν (Ai1··· ,im) , define

Iνi1··· ,im, j ≡

[α +

j−1

∑k=1

ν(Ai1··· ,im,k

),α +

j

∑k=1

ν(Ai1··· ,im,k

))