Lemma 58.1.9 on Page 6229 makes possible a definition of convolution of two probability
measures defined on ℬ
(E)
where E is a separable Banach space as well as some other
interesting theorems which held earlier in the context of locally compact spaces. I will
first show a little theorem about density of continuous functions in L^{p}
(E)
and then
define the convolution of two finite measures. First here is a simple technical
lemma.
Lemma 58.14.1Suppose K is a compact subset of U an open set in E a metric space.Then there exists δ > 0 such that
( C)
dist(x,K )+ dist x,U ≥ δ for all x ∈ E.
Proof: For each x ∈ K, there exists a ball, B
(x,δx)
such that B
(x,3δx)
⊆ U.
Finitely many of these balls cover K because K is compact, say
{B (xi,δxi)}
_{i=1}^{m}.
Let
0 < δ < min(δ : i = 1,2,⋅⋅⋅,m ).
xi
Now pick any x ∈ K. Then x ∈ B
(x ,δ )
i xi
for some x_{i} and so B
(x,δ)
⊆ B
(x ,2δ )
i xi
⊆ U.
Therefore, for any x ∈ K,dist
(x,U C)
≥ δ. If x ∈ B
(x,2δ )
i xi
for some x_{i}, it follows
dist
( C )
x,U
≥ δ because then B
(x,δ)
⊆ B
(xi,3δxi)
⊆ U. If x
∈∕
B
(xi,2δxi)
for any of the
x_{i}, then x
∕∈
B
(y,δ)
for any y ∈ K because all these sets are contained in some
B
(xi,2δxi)
. Consequently dist
(x,K )
≥ δ. This proves the lemma.
From this lemma, there is an easy corollary.
Corollary 58.14.2Suppose K is a compact subset of U, an open set in E ametric space. Then there exists a uniformly continuous function f defined on all ofE, having values in
[0,1]
such that f
(x)
= 0 if x
∕∈
U and f
(x)
= 1 if x ∈ K.
Proof: Consider
( )
-----dist-x,UC-------
f (x) ≡ dist(x,U C)+ dist(x,K ).
where δ is the constant of Lemma 58.14.1. Now it is a general fact that
|dist(x,S)− dist(x′,S)| ≤ d(x,x′).
Therefore,
2
|f (x)− f (x′)| ≤-d(x,x′)
δ
and this proves the corollary.
Now suppose μ is a finite measure defined on the Borel sets of a separable Banach
space, E. It was shown above that μ is inner and outer regular. Lemma 58.1.9 on Page
6229 shows that μ is inner regular in the usual sense with respect to compact sets. This
makes possible the following theorem.
Theorem 58.14.3Let μ be a finite measure on ℬ
(E )
where E is a separable Banachspace and let f ∈ L^{p}
(E; μ)
. Then for any ε > 0, there exists a uniformly continuous,bounded g defined on E such that
||f − g||Lp(E) < ε.
Proof: As usual in such situations, it suffices to consider only f ≥ 0. Then by
Theorem 9.3.9 on Page 644 and an application of the monotone convergence theorem,
there exists a simple measurable function,
m
s (x) ≡ ∑ cX (x)
k=1 k Ak
such that
||f − s||
_{Lp(E)
}< ε∕2. Now by regularity of μ there exist compact sets, K_{k} and
open sets, V_{k} such that 2∑_{k=1}^{m}
|ck|
μ
(Vk ∖K )
^{1∕p}< ε∕2 and by Corollary 58.14.2 there
exist uniformly continuous functions g_{k} having values in
[0,1]
such that g_{k} = 1 on K_{k}
and 0 on V _{k}^{C}. Then consider
∑m
g(x) = ckgk(x).
k=1
This function is bounded and uniformly continuous. Furthermore,
( ∫ | |p )1∕p
||∑m ∑m ||
||s − g||Lp(E) ≤ E|| ckXAk (x) − ckgk(x)|| dμ
( (k=1 k=1 ) )1∕p
∫ m∑ p
≤ |ck||XAk (x)− gk(x)|
E k(=1∫ )
∑m p 1∕p
≤ |ck| E|XAk (x )− gk (x )| dμ
k=1 ( )1∕p
∑m ∫ p
≤ |ck| V ∖K 2 dμ
k=1m k k
∑ 1∕p
= 2 |ck|μ(Vk ∖ K) < ε∕2.
k=1
and so ∪_{i=1}^{∞}A_{i}∈G because the above is the sum of Borel measurable functions. By the
lemma on π systems, Lemma 10.12.3 on Page 923, it follows G = σ
(K )
= ℬ
(E )
.
Similarly, x → μ
( ∑m )
A − j=1xj
is also Borel measurable whenever A ∈ℬ
(E )
. Finally
note that
ℬ (E )× ⋅⋅⋅×ℬ (E)
contains the open sets of E^{m} because the separability of E implies the existence of a
countable basis for the topology of E^{m} consisting of sets of the form ∏_{i=1}^{m}U_{i} where the
U_{i} come from a countable basis for E. Since every open set is the countable union of
sets like the above, each being a measurable box, the open sets are contained
in
ℬ (E )× ⋅⋅⋅×ℬ (E)
which implies ℬ
m
(E )
⊆ℬ
(E )
×
⋅⋅⋅
×ℬ
(E)
also. This proves the lemma.
With this lemma, it is possible to define the convolution of two finite measures.
Definition 58.14.5Let μ and ν be two finite measures on ℬ
(E )
, for E a separableBanach space. Then define a new measure, μ ∗ ν on ℬ
(E )
as follows
∫
μ ∗ν(A ) ≡ ν(A− x) dμ(x).
E
This is well defined because of Lemma 58.14.4which says that x → ν
(A − x)
is Borelmeasurable.
Here is an interesting theorem about convolutions. However, first here is a little
lemma. The following picture is descriptive of the set described in the following
lemma.
PICT
Lemma 58.14.6For A a Borel set in E, a separable Banach space, define
SA ≡ {(x,y) ∈ E × E : x + y ∈ A}
Then S_{A}∈ℬ
(E )
×ℬ
(E )
, the σ algebra of product measurable sets, the smallestσ algebra which contains all the sets of the form A × B where A and B areBorel.
Proof: Let K denote the open sets in E. Then K is a π system. Let
G ≡ {A ∈ σ(K) = ℬ(E ) : SA ∈ ℬ(E) × ℬ(E)}.
Then K⊆G because if U ∈K then S_{U} is an open set in E × E and all open sets are in
ℬ
(E)
×ℬ
(E )
because a countable basis for the topology of E × E are sets of the form
B × C where B and C come from a countable basis for E. Therefore, K⊆G. Now let
A ∈G. For
(x,y)
∈ E × E, either x + y ∈ A or x + y
∕∈
A. Hence E × E = S_{A}∪ S_{AC}
which shows that if A ∈G then so is A^{C}. Finally if
{A }
i
is a sequence of disjoint sets of
G
S ∪∞i=1Ai = ∪∞i=1SAi
and this shows that G is also closed with respect to countable unions of disjoint sets.
Therefore, by the lemma on π systems, Lemma 10.12.3 on Page 923 it follows
G = σ
(K )
= ℬ
(E )
. This proves the lemma.
Theorem 58.14.7Let μ, ν, and λ be finite measures on ℬ
(E )
for E a separableBanach space. Then
μ ∗ν = ν ∗μ (58.14.23)
(58.14.23)
(μ ∗ν)∗ λ = μ ∗(ν ∗λ) (58.14.24)
(58.14.24)
If μ is the distribution for an E valued random variable, X and if ν is the distribution foran E valued random variable, Y, and X and Y are independent, then μ ∗ ν is thedistribution for the random variable, X + Y . Also the characteristic function of aconvolution equals the product of the characteristic functions.
∫
ϕ(μ∗ν)(t∗) = eit∗(z)d(μ ∗ν)(z)
∫E ∫
= eit∗(x+y)dν(y)dμ(x)
E E
∫ ∫ it∗(x) it∗(y)
= E E e e dν(y)dμ(x)
( ∫ ∗ ) (∫ ∗ )
= eit(y)dν (y) eit (x)dμ(x)
(E ) ( E)
= E eit∗(X) E eit∗(Y)
Since ϕ_{λ}
∗
(t )
= ϕ_{(μ∗ν)
}
∗
(t )
, it follows λ = μ ∗ ν.
Note the last part of this argument shows the characteristic function of a
convolution equals the product of the characteristic functions. This proves the
theorem.