60.5 Bochner’s Theorem In Infinite Dimensions
Let X be a real vector space and let X∗ denote the space of real valued linear mappings
defined on X. Then you can consider each x ∈ X as a linear transformation
defined on X∗ by the convention x∗→ x∗
Now let Λ be a Hamel basis. For a
description of what one of these is, see Page ??
. It is just the usual notion of a basis.
Thus every vector of X
is a finite linear combination of vectors of Λ in a unique
Now consider ℝΛ the space of all mappings from Λ to ℝ. In different notation, this is
of the form
Since Λ is a Hamel basis, there exists a one to one and onto mapping, θ : X∗→ ℝΛ
Now denote by σ
algebra of sets of X∗
such that each x
measurable with respect to this σ
whenever B is a Borel set in ℝ.
Let ℰ denote the algebra of disjoint unions of sets of ℝΛ of the form
where Ay = ℝ except for finitely many y.
Lemma 60.5.1 Let A denote sets of the form
where U ∈ℰ. Then A is an algebra and σ
Proof: Since ℰ is an algebra it is clear A is also an algebra. Also, A⊆ σ
you could let
have only one Ay
not equal to ℝ
and all the others equal to ℝ
I need to verify that for an arbitrary x,
it is measurable with
respect to σ
However, this is true because if x
is arbitrary, it is a linear combination
some finite set of functions in Λ and so, x
being a linear combination of
measurable functions implies it is itself measurable.
By definition, θ−1
whenever U ∈ℰ
. Now let G
denote those sets, U
is a σ
algebra which contains ℰ
. This proves the last claim. This proves the lemma.
Definition 60.5.2 Let ψ : X → ℂ. Then ψ is said to be pseudo continuous if whenever
is a finite subset of X and a
is continuous. ψ is said to be positive definite if
ψ is said to be a characteristic function if there exists a probability measure, μ defined on
Note that x∗→ eix∗
Using Kolmogorov’s extension theorem on Page 58.2.3, there exists a generalization of
Bochner’s theorem found in [?]. For convenience, here is Kolmogorov’s theorem.
Theorem 60.5.3 (Kolmogorov extension theorem) For each finite set
suppose there exists a Borel probability measure, νJ = νt1
tn defined on the Borel sets of
t∈JMt for Mt = ℝnt for nt an integer, such that the following consistency condition
where if si = tj, then Gsi = Ftj and if si is not equal to any of the indices, tk, then
Gsi = Msi. Then for ℰ defined as in Definition 12.4.1, adjusted so that ±∞ never
appears as any endpoint of any interval, there exists a probability measure, P and a σ
algebra ℱ = σ
is a probability space. Also there exist measurable functions, Xs : ∏
t∈IMt → Ms defined
for each s ∈ I such that for each
where Ft = Mt for every t
and Fti is a Borel set. Also if f is a
nonnegative function of finitely many variables, xt1,
,xtn, measurable with respect to
, then f is also measurable with respect to ℱ and
Theorem 60.5.4 Let X be a real vector space and let X∗ be the space of linear
functionals defined on X. Also let ψ : X → ℂ. Then ψ is a characteristic function
if and only if ψ
and ψ is pseudo continuous at 0.
Proof: Suppose first ψ is a characteristic function as just described. I need to show it
is positive definite and pseudo continuous. It is obvious ψ
= 1 in this case.
and this is obviously a continuous function of a by the dominated convergence theorem.
It only remains to verify the function is positive definite. However,
as in the earlier discussion of what it means to be positive definite given on Page
Next suppose the conditions hold. Define for t ∈ ℝn and
is continuous at 0, equals 1 there, and is positive definite. It follows from
Bochner’s theorem, Theorem 58.21.7 on Page 6539 there exists a measure μ
defined on the Borel sets of ℝn such that
I need to verify the measures are consistent to use Kolmogorov’s theorem. Specifically,
I need to show
and so, by uniqueness of characteristic functions,
which verifies the necessary consistency condition for Kolmogorov’s theorem.
It follows there exists a probability measure μ defined on σ
and random variables,
for each y ∈
Λ such that whenever
where Ay = ℝ whenever y
. This defines a measure on
of sets of
By Lemma 60.5.1 it follows
Thus ν is a measure because μ is and θ is one to one.
I need to check whether ν works. Let x = ∑
k=1mtkyk and let a typical element of ℝΛ
be denoted by z. Then by Kolmogorov’s theorem above,
where the last equality comes from 60.5.16. Since Λ is a Hamel basis, it follows that for
every x ∈ X,
This proves the theorem.