RANDOM FIELDS AND THEIR GEOMETRY

44 2. Gaussian fields
Corollary 2.1.4 Under the conditions of Theorem 2.1.3 there exists a universal
constant K such that
(2.1.8)
E {ωf,d(δ)} ≤ K
� δ
0
H 1/2 (ε) dε.
Note that this is not quite enough to establish the a.s. continuity of
f. Continuity is, however, not far away, since the same construction used
to prove Theorem 2.1.3 will also give us the following, which, with the
elementary tools we have at hand at the moment 6 , neither follows from,
nor directly implies, (2.1.8).
Theorem 2.1.5 Under the conditions of Theorem 2.1.3 there exists a random
η ∈ (0, ∞) and a universal constant K such that
(2.1.9)
for all δ < η.
ωf,d(δ) ≤ K
� δ
0
H 1/2 (ε) dε,
Note that (2.1.9) is expressed in terms of the d modulus of continuity.
Translating this to a result for the τ modulus is trivial.
We shall see later that if f is stationary then the convergence of the
entropy integral is also necessary for continuity and that continuity and
boundedness always occur together (Theorem 2.6.4). Now, however, we
shall prove Theorems 2.1.3 and 2.1.5 following the approach of Talagrand
[92]. The original proof of Theorem 2.1.5 is due to Dudley [26], and, in
fact, things have not really changed very much since then. Immediately
following the proofs, in Section 2.2, we shall look at examples, to see how
entropy arguments work in practice. You may want to skip to the examples
before going through the proofs first time around.
We start with the following almost trivial, but important, observations.
Observation 2.1.6 If f is a separable process on T then sup t∈T ft is a
well defined (i.e. measurable) random variable.
Measurability follows directly from Definition 1.1.3 of separability which
gave us a countable dense subset D ⊂ T for which
sup
t∈T
ft = sup ft.
t∈D
The supremum of a countable set of measurable random variables is always
measurable.
One can actually manage without separability for the rest of this Section,
in which case
sup
�
E
�
sup ft
t∈F
�
�
: F ⊂ T, F finite
6 See, however Theorem ?? below, to see what one can do with better tools.