61.4. A MAJOR EXISTENCE AND CONVERGENCE THEOREM 1999

Since there are only finitely many sets in the union, there exists s≤ k−1 such that every ballabout x contains points of Cri

s but from 61.4.10, every ball about x contains points of(Cri

s)C

which implies x ∈ ∂Cris ⊆ F by induction. It is not possible that d (x,ak)> ri and yet have

x in ∂Crik . This follows from the description in 61.4.10. If d (x,ak) = ri then by definition,

x ∈ F. The only other case to consider is that x /∈ int(B(ak,ri)∩C) but x ∈ B(ak,ri). From61.4.10, every ball about x contains points of C. However, since x ∈ B(ak,ri) , a smallenough ball is contained in B(ak,ri) . Therefore, every ball about x must also contain pointsof CC since otherwise, x ∈ int(B(ak,ri)∩C) . Thus x ∈ ∂C⊆ F by assumption. Now applywhat was just shown to the case where C = E, the whole space. In this case, ∂E ⊆ Fbecause ∂E = /0. Then keep applying what was just shown to the Ai1,··· ,in . This proves theclaim.

From the claim, ν (int(Ai1,··· ,in)) = ν (Ai1,··· ,in) whenever ν = µ or µn. This is becausethat in Ai1,··· ,in which is not in int(Ai1,··· ,in) is in F which has measure zero.

Some functions on [0,1)

By the axiom of choice, there exists xi1,··· ,im ∈ int(Ai1,··· ,im) whenever

int(Ai1,··· ,im) ̸= /0.

For ν = µn or µ, define the following functions. For ω ∈ Iνi1,··· ,im

Zνm (ω)≡ xi1,··· ,im .

This defines the functions, Zµnm and Zµ

m. Note these functions have the same values but onslightly different intervals. Here is an important claim.

Claim 2 (Limit on µn): For a.e. ω ∈ [0,1), limn→∞ Zµnm (ω) = Zµ

m (ω) .Proof of the claim: This follows from the weak convergence of µn to µ and Lemma

61.4.1. This lemma implies µn (int(Ai1,··· ,im))→ µ (int(Ai1,··· ,im)) . Thus by the construc-tion described above, µn (Ai1,··· ,im)→ µ (Ai1,··· ,im) because of claim 1 and the construction

of F in which it is always a set of measure zero. It follows that if ω ∈ int(

Iµ

i1,··· ,im

), then

for all n large enough, ω ∈ int(

Iµni1,··· ,im

)and so Zµn

m (ω) = Zµm (ω) . Note this convergence

is very far from being uniform.Claim 3 (Limit on size of sets, fixed measure): For ν = µn or µ,{Zν

m}∞

m=1 is uniformlyCauchy independent of ν .

Proof of the claim: For ω ∈ Iνi1,··· ,im , then by the construction,

ω ∈ Iνi1,··· ,im,im+1··· ,in

for some im+1 · · · , in. Therefore, Zνm (ω) and Zν

n (ω) are both contained in Ai1,··· ,im whichis contained in B(aim ,rm) . Since ω ∈ [0,1) was arbitrary, and rm → 0, it follows thesefunctions are uniformly Cauchy as claimed.

Let Xν (ω) = limm→∞ Zνm (ω). Since each Zν

m is continuous off a set of measure zero, itfollows from the uniform convergence that Xν is also continuous off a set of measure zero.

Claim 4: For a.e. ω,limn→∞

X µn (ω) = X µ (ω) .