GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS 13

Now, since N is |n| x Lebesgue-null, it follows that for |n|-a.e. s

and every £, and so certainly for |n| x Lebesgue-a.e. (s,£) the set

X ' [iirs'-£] is Lebesgue-null. But it is well known that the set of

Wiener paths x whose value at a particular time s lies in a Lebesgue-null

set is a set of m-measure zero [50, Theorem 29.1, p. 437]. Hence, by the

Fubini theorem, H~ (N) is |n| * m x Lebesgue-null. Thus (*) is estab-

lished.

It now follows from (*) that for every X 0 and m x Lebesgue-a.e.

(x,5),

/

|e(S)x_1/2x(s)+0|d|n|(s)

£ / l|eCs,-)|Ld|n|(s) = liell , + -•

(0,t) (0,t) ° ° ' " ' '~l;n

Hence for every X 0 and m x Lebesgue-a.e. (x,C)

F,(X~1/2x+5) = / e(s,x"1/2x(s)+Odn(s)

L (0,t)

is defined and we have

\I,(\'1,2x+Z)\

_ / |e(s,x"1/2x(s)+0|d|n|(s)

1

(0,t)

(0.16) . / ||e(sf.)ILd|n|(8) = I I e ( |

—

-, .

(0,t) ~1'"

This concludes the proof of Lemma 0.1. •

We remark that even if e is everywhere defined, there may be contin-

uous functions y for which F,(y) fails to be defined. If e is everywhere

defined and bounded, then F,(y) is defined for all y in C[0,t]. Note

that even when 6 is essentially bounded, the graph of the function

s H x"1/2x(s)+£ could lie in {(s,v) e (0,t) x RN: |e(s,v)| | | e (s, •)11J

for some X 0, x and £. In this case, (0.15) could fail since the

second inequality in (0.16) could fail.

Our second lemma does not seem to be explicitly stated in the stan-

dard references on the Bochner integral. We include its easy proof.

LEMMA 0.2 (Bochner integrals depending on a parameter). Let E be a

complex Banach space. Let (A,y) be a a-finite measure space and let T

be a metric space. Consider the function g : T x A + E [or jt(E)].