60.9 Abstract Wiener Spaces
This material follows [?], [?] and [?]. More can be found on this subject in these
references. Here H will be a separable real Hilbert space.
Definition 60.9.1 Cylinder sets in H are of the form
where F ∈ℬ
, the Borel sets of ℝn and
are given vectors in H. Denote this
collection of cylinder sets as C.
Proposition 60.9.2 The cylinder sets form an algebra of sets.
Proof: First note the complement of a cylinder set is a cylinder set.
Now consider the intersection of two of these cylinder sets. Let the cylinder sets be
The first of these equals
and the second equals
Therefore, their intersection equals
a cylinder set.
Now it is clear the whole of H and ∅ are cylinder sets given by
respectively and so if C1,C2 are two cylinder sets,
which was just shown to be a cylinder set. Hence
a cylinder set. This proves the proposition.
It is good to have a more geometrical description of cylinder sets. Letting A be
a cylinder set as just described, let P denote the orthogonal projection onto
Also let α
: PH → ℝn
be given by
This is continuous but might not be one to one if the ei are not a basis for example. Then
those x ∈ PH
For any x ∈ H,
for each k and so
Thus Px ∈ α−1
which is a Borel set of PH
so the cylinder set is contained in
which is of the form
On the other hand, consider a set of the form
where G is a Borel set in PH. There is a basis for PH consisting of a subset of
For simplicity, suppose it is
. Then let
: PH → ℝk
Thus α is a homeomorphism of PH and ℝk so α1
is a Borel set of
is a Borel set of ℝn.
This has proved the following important
Proposition illustrated by the following picture.
Proposition 60.9.3 The cylinder sets are sets of the form
where M is a finite dimensional subspace and B is a Borel subset of M. Furthermore, the
collection of cylinder sets is an algebra.
Lemma 60.9.4 σ
, the smallest σ algebra containing C, contains the Borel sets
Proof: It follows from the definition of these cylinder sets that if fi
that fi ∈ H′,
then with respect to σ
is measurable. It follows that every
linear combination of the fi
is also measurable with respect to σ
. However, this set of
linear combinations is dense in
and so the conclusion of the lemma follows from
on Page 6264
. This proves the lemma.
Also note that the mapping
is a σ
measurable map. Restricting it to
it is Borel measurable.
Next is a definition of a Gaussian measure defined on C.
While this is what it is called, it
is a fake measure in general because it cannot be extended to a countably additive
measure on σ
. This will be shown below.
Definition 60.9.5 Let Q ∈ℒ
be self adjoint and satisfy
for all x ∈ H,x≠0. Define ν on the cylinder sets, C by the following rule. For
an orthonormal set in H, where here
Note that the cylinder set is of the form
Thus if B + M⊥ is a typical cylinder set, choose an orthonormal basis for M,
and do the above definition with F
To see this last claim which is like what was done earlier, let
To see the other inclusion, if t ∈ F and y ∈ span
then if x
= 0 for all
and so x
Lemma 60.9.6 The above definition is well defined.
be another orthonormal set such that for
Borel sets in ℝn,
I need to verify ν
is the same using either
a ∈ G.
Therefore, for this x
it is also true that
other words for a ∈ G,
Let L ∈ℒ
be defined by
are orthonormal, this mapping is unitary. Also this has shown
where L∗ has the ijth entry Lij∗ =
as above and
is the inverse of L
is unitary. Thus
showing that LG = F and L∗F = G.
Now let θet ≡∑
itiei with θf defined similarly. Then the definition of ν
Now change the variables letting t = Ls where s ∈ G.
From the definition,
where from the definition,
and so θe∗θe
is the identity on ℝn
and similar reasoning yields θeθe∗
identity on θe
. Then using the change of variables formula and the fact
where part of the justification is as follows.
This proves the lemma.
It would be natural to try to extend ν to the σ algebra determined by C and obtain a
measure defined on this σ algebra. However, this is always impossible in the case where
Q = I.
Proposition 60.9.7 For Q = I, ν cannot be extended to a measure defined on
whenever H is infinite dimensional.
be a complete orthonormal set of vectors in
Then first note that
is a cylinder set.
However, H is also equal to the countable union of the sets,
where an →∞.
Now pick an
so large that the above is smaller than 1∕
. This can be done because for
no matter what choice of n,