6

DAVID DOUGLAS ENGEL

g U - , . . . ^ ) d X ( 0 - * . d X ( t )

1 n 1 n

0 t

n

o - -

t

T

— 1 n—

= L . I . M .

k • °°

| g

k

( t

1

, . . . , t

n

) d X ( t

1

) - . . d X ( t

n

) (3)

0t •••t T

— 1 n

where g (t , . . . , t ) are step functions in S(Q ) converging to

g ( t - , . . . , t ) in L (3R )-mean. Note that if g ( t , . . . , t ) i s the character-

1 n I n

i s t i c function of the set I x»--x i where I . I . , - then

I n l l+ l

••• I g O : . , . . . ^ )dX(t

1

)---dX(t ) = X ( L ) - . . X ( 1 ).

J J l n l n 1 n

0t ••• t T

— 1 n —

This defines a f i n i t e l y additive set function X (B) defined on the set

of a l l elementary subsets B of S = { ( t , , . . . , t ) I 0t «*«t T} whose

1 n — 1 n

2

values are in the space L (ft). The fact that the integral (3) exists for

all geS(Q ) n L (R ) implies that the finitely additive set function

X has an extension to a countably additive set function X (B) defined

for all Borel subsets ECS.

The connection between the homogeneous chaos Q and the multiple

Wiener integral is given by the following Theorem and its Corollary.

THEOREM 1.2 If X(t) = X(t,oo) is standard Brownian Motion, then

dX(t-)---dX(tn) = H (X(T),T).

1 n

0t-«*• t T

1 n

Remark: This says that the "measure" of the set S given by X is

exactly H (X(T),T),