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),
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