RANDOM FIELDS AND THEIR GEOMETRY

10 1. Random fields
FIGURE 1.3.2. Construction of some lower layers.
The class of lower layers in [0, 1] 2 certainly includes all sets made up
by taking those points that lie between the axes and one of the step-like
structures of Figure 1.3, where each step comes from the horizontal and
vertical sides of some Tnj with, perhaps, different n.
Note that since the squares Cnj are disjoint for all n and j, the random
variables W (Cnj) are independent. Also |Cnj| = 4 −(n+1) for all n, j.
Let D be the negative diagonal {(s, t) ∈ [0, 1] 2 : s + t = 1} and Lnj =
D ∩ Tnj. For each n ≥ 1, each point p = (s, t) ∈ D belongs to exactly one
such interval L n,j(n,p) for some unique j(n, p).
For each p ∈ D and M < ∞ the events
Enp
∆
= {W (C n,j(n,p)) > M2 −(n+1) }
are independent for n = 0, 1, 2, ..., and since W (Cnj)/2 −(n+1) is standard
normal for all n and j they also have the same positive probability. Thus,
for each p we have that, for almost all ω, the events Enp occur for all but
finitely many n. Let n(p) = n(p, ω) be the least such n.
Since the events Enp(ω) are measurable jointly in p and ω, Fubini’s theorem
implies that, with probability one, for almost all p ∈ D (with respect
to Lebesgue measure on D) some Enp occurs, and n(p) < ∞. Let
Vω = �
Aω
Bω
p∈D
T n(p),j(n(p),p),
∆
= {(s, t) : s + t ≤ 1} ∪ Vω,
∆
= Aω \ �
Cn(p),j(n(p),p). p∈D