61.4. A MAJOR EXISTENCE AND CONVERGENCE THEOREM 2001

Since ε > 0 is arbitrary, this shows m((Xν)−1 (∂Ai1,··· ,im)

)= 0.

If ω ∈ Iνi1,··· ,im , then from the construction, Zν

p (ω) ∈ int(Ai1,··· ,im) for all p ≥ k. There-fore, taking a limit, as p→ ∞,

Xν (ω) ∈ int(Ai1,··· ,im)∪∂Ai1,··· ,im

and soIνi1,··· ,im ⊆ (Xν)

−1(int(Ai1,··· ,im)∪∂Ai1,··· ,im)

but also, if Xν (ω) ∈ int(Ai1,··· ,im) , then Zνp (ω) ∈ int(Ai1,··· ,im) for all p large enough and

so

(Xν)−1

(int(Ai1,··· ,im))

⊆ Iνi1,··· ,im ⊆ (Xν)

−1(int(Ai1,··· ,im)∪∂Ai1,··· ,im)

Therefore,

m((Xν)

−1(int(Ai1,··· ,im))

)≤ m

(Iνi1,··· ,im

)≤ m

((Xν)

−1(int(Ai1,··· ,im))

)+m

((Xν)

−1(∂Ai1,··· ,im)

)= m

((Xν)

−1(int(Ai1,··· ,im))

)which shows

m((Xν)

−1(int(Ai1,··· ,im))

)= m

(Iνi1,··· ,im

)= ν (Ai1,··· ,im) . (61.4.11)

Also

m((Xν)

−1(int(Ai1,··· ,im))

)≤ m

((Xν)

−1(Ai1,··· ,im)

)≤ m

((Xν)

−1(int(Ai1,··· ,im)∪∂Ai1,··· ,im)

)= m

((Xν)

−1(int(Ai1,··· ,im))

)Hence from 61.4.11,

ν (Ai1,··· ,im) = m((Xν)

−1(int(Ai1,··· ,im))

)= m

((Xν)

−1(Ai1,··· ,im)

)(61.4.12)

Now let U be an open set in E. Then letting

Hk ={

x ∈U : dist(x,UC)≥ rk

}