RANDOM FIELDS AND THEIR GEOMETRY

2 1. Random fields
Definition 1.1.2 Let (Ω, F, P) be a complete probability space and T a
topological space. Then a measurable mapping f : Ω → R T is called a real
valued random field 1 . Measurable mappings from Ω to (R T ) d , d > 1, are
called vector valued random fields. If T ⊂ R N , we call f an (N, d) random
field, and if d = 1 simply an N-dimensional random field.
We shall generally not distinguish between
ft ≡ f(t) ≡ f(t, ω) ≡ (f(ω))(t),
etc., unless there is some special need to do so. Throughout, we shall demand
that all random fields are separable, a property due originally to
Doob [22], which implies conditions on both T and X.
Definition 1.1.3 An R d -valued random field f, on a topological space T ,
is called separable if there exists a countable dense subset D ⊂ T and a
fixed event N with P{N} = 0 such that, for any closed B ⊂ R d and open
I ⊂ T ,
{ω : f(t, ω) ∈ B, ∀t ∈ I} ∆ {ω : f(t, ω) ∈ B, ∀t ∈ I ∩ D} ⊂ N.
Here ∆ denotes the usual symmetric difference operator, so that
(1.1.2)
A∆B = (A ∩ B c ) ∪ (A c ∩ B),
where A c is the complement of A.
Since you have read the Preface, you also know that most of this book
centres on Gaussian random fields. The next section is devoted to defining
these and giving some of their basic properties. Fortunately, most of these
have little to do with the specific geometric structure of the parameter
space T , and after decades of polishing even proofs gain little in the way
of simplification by restricting to special cases such as T = R N . Thus, at
least for a while, we can and shall work in as wide as possible generality.
Only when we get to geometry, in Chapter 3, will we need to specialise,
either to Euclidean T or to Riemannian manifolds.
1.2 Gaussian fields
The centrality of Gaussian fields to this book is due to two basic factors:
• Gaussian processes have a rich, detailed and very well understood
general theory, which makes them beloved by theoreticians.
1 On notation: While we shall follow the standard convention of denoting random
variables by upper case Latin characters, we shall use lower case to denote random
functions. The reason for this will be become clear in Chapter 3, where we shall need
the former for tangent vectors.