756 CHAPTER 27. ANALYTICAL CONSIDERATIONS

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 SU is an open set in E×E and all open sets are inB (E)×B (E) because a countable basis for the topology of E ×E are sets of the formB×C where B and C come from a countable basis for E. Therefore, K ⊆ G . Now letA ∈ G . For (x,y) ∈ E×E, either x+ y ∈ A or x+ y /∈ A. Hence E×E = SA ∪ SAC whichshows that if A ∈ G then so is AC. Finally if {Ai} 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. There-fore, by the lemma on π systems, Lemma 9.3.2 on Page 243 it follows G = σ (K ) =B (E) . This proves the lemma.

Theorem 27.7.7 Let µ, ν , and λ be finite measures on B (E) for E a separableBanach space. Then

µ ∗ν = ν ∗µ (27.12)

(µ ∗ν)∗λ = µ ∗ (ν ∗λ ) (27.13)

If µ is the distribution for an E valued random variable, X and if ν is the distribution for anE valued random variable, Y, and X and Y are independent, then µ ∗ν is the distributionfor the random variable, X +Y . Also the characteristic function of a convolution equalsthe product of the characteristic functions.

Proof: First consider 27.12. Letting A ∈B (E) , the following computation holds fromFubini’s theorem and Lemma 27.7.6

µ ∗ν (A) ≡∫

Eν (A− x)dµ (x) =

∫E

∫E

XSA (x,y)dν (y)dµ (x)

=∫

E

∫E

XSA (x,y)dµ (x)dν (y) = ν ∗µ (A) .

Next consider 27.13. Using 27.12 whenever convenient,

(µ ∗ν)∗λ (A) ≡∫

E(µ ∗ν)(A− x)dλ (x)

=∫

E

∫E

ν (A− x− y)dµ (y)dλ (x)

while

µ ∗ (ν ∗λ )(A) ≡∫

E(ν ∗λ )(A− y)dµ (y)

=∫

E

∫E

ν (A− y− x)dλ (x)dµ (y)

=∫

E

∫E

ν (A− y− x)dµ (y)dλ (x) .

The necessary product measurability comes from Lemma 27.7.4.Recall