PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

154 Geoffrey Grimmett
Fig. 2.2. An open cluster, surrounded by a closed circuit in the dual.
2.4 A Partial Order
There is a natural partial order on Ω, namely ω1 ≤ ω2 if and only if ω1(e) ≤
ω2(e) for all e. This partial order allows us to discuss orderings of probability
measures on (Ω, F). We call a random variable X on (Ω, F) increasing if
X(ω1) ≤ X(ω2) whenever ω1 ≤ ω2,
and decreasing if −X is increasing. We call an event A (i.e., a set in F)
increasing (resp. decreasing) if its indicator function 1A, given by
�
1 if ω ∈ A,
1A(ω) =
0 if ω /∈ A,
is increasing (resp. decreasing).
Given two probability measures µ1 and µ2 on (Ω, F) we say that µ1 dominates
µ2, written µ1 ≥ µ2, if µ1(A) ≥ µ2(A) for all increasing events A.
Using this partial order on measures, it may easily be seen that the probability
measure Pp is non-decreasing in p, which is to say that
(2.4) Pp1 ≥ Pp2 if p1 ≥ p2.
General sufficient conditions for such an inequality have been provided by
Holley [193] and others (see Holley’s inequality, Theorem 5.5), but there is
a simple direct proof in the case of product measures. It makes use of the
following elementary device.