27.7. CONVOLUTION AND SUMS 755

I just showed G ⊇K . Now suppose A ∈ G . Then

µ

(AC +

m

∑i=1

yi

)= µ (E)−µ

(A+

m

∑i=1

yi

)and so AC ∈ G whenever A ∈ G . Next suppose {Ai} is a sequence of disjoint sets of G .Then

µ

((∪∞

i=1Ai)+m

∑j=1

y j

)= µ

(∪∞

i=1

(Ai +

m

∑j=1

y j

))=

∞

∑i=1

µ

(Ai +

m

∑j=1

y j

)and so ∪∞

i=1Ai ∈ G because the above is the sum of Borel measurable functions. By thelemma on π systems, Lemma 9.3.2 on Page 243, it follows G =σ (K )=B (E) . Similarly,x→ µ

(A−∑

mj=1 x j

)is also Borel measurable whenever A ∈B (E). Finally note that

B (E)×·· ·×B (E)

contains the open sets of Em because the separability of E implies the existence of a count-able basis for the topology of Em consisting of sets of the form ∏

mi=1 Ui where the Ui come

from a countable basis for E. Since every open set is the countable union of sets like theabove, each being a measurable box, the open sets are contained in B (E)× ·· ·×B (E)which implies B (Em)⊆B (E)×·· ·×B (E) also. ■

With this lemma, it is possible to define the convolution of two finite measures.

Definition 27.7.5 Let 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)≡∫

Eν (A− x)dµ (x) .

This is well defined because of Lemma 27.7.4 which says that x→ ν (A− x) is Borel mea-surable.

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.

AE

E

SA

Lemma 27.7.6 For A a Borel set in E, a separable Banach space, define

SA ≡ {(x,y) ∈ E×E : x+ y ∈ A}

Then SA ∈B (E)×B (E) , the σ algebra of product measurable sets, the smallest σ alge-bra which contains all the sets of the form A×B where A and B are Borel.