61.9. ABSTRACT WIENER SPACES 2031

Thus Px ∈ α−1 (F) , which is a Borel set of PH and

x = Px+(I−P)x

so the cylinder set is contained in

α−1 (F)+(PH)⊥

which is of the form(Borel set of PH)+(PH)⊥

On the other hand, consider a set of the form

G+(PH)⊥

where G is a Borel set in PH. There is a basis for PH consisting of a subset of {e1 · · · ,en} .For simplicity, suppose it is {e1 · · · ,ek}. Then let α1 : PH→ Rk be given by

α1 (x)≡ ((x,e1) , · · · ,(x,ek))

Thus α is a homeomorphism of PH and Rk so α1 (G) is a Borel set of Rk. Now

α−1(

α1 (G)×Rn−k)= G

and α1 (G)×Rn−k is a Borel set ofRn. This has proved the following important Propositionillustrated by the following picture.

B

B+M⊥

Proposition 61.9.3 The cylinder sets are sets of the form

B+M⊥

where M is a finite dimensional subspace and B is a Borel subset of M. Furthermore, thecollection of cylinder sets is an algebra.

Lemma 61.9.4 σ (C ) , the smallest σ algebra containing C , contains the Borel sets ofH,B (H).

Proof: It follows from the definition of these cylinder sets that if fi (x) ≡ (x,ei) , sothat fi ∈ H ′, then with respect to σ (C ) , each fi is measurable. It follows that every linearcombination of the fi is also measurable with respect to σ (C ). However, this set of linear