R.H.CAMERON AND D.A. STORVICK

1 1 m-1 2

n

1 ,, . 1

J v ( t ) d x ( t ) - x(l ) I v ( s ) d s + E E J x r

}

( t ) d x ( t ) J x r

;

( s ) v ( s ) d s

0 m 0 n=0 k=l 0 n 0 n

f v(s)[x(l) + V E

X

(k)(s)

/V

k)

(t)dx(t)]ds

1 m-1

2n

/ t N

1

0 n=0 k=l n O n

1

= J v(s)x

m

(s)ds .

Thus, we have

1 1

(2.6) J v (t)dx(t ) = J v(s)x (s)ds .

0 0

We now show fo r x e c[0,1] tha t

d[x] (s)

A

- a l - -

*ms

for s e [0,1] , s / a + - ~ - , k = 0 , 1 , 2 , . . . ,2 m .

Now by Remark 2 ,

x

m

(s) = x(l ) + ^ E

X

( k )

( t ) J X

( k )

(s)dx(s )

n=0 k=l n O n

m-1 2

n

/ 1 X

1

(1) + E £ X ( k ) ( t ) J X ( k ) (s)d[x ] (s )

n=0 k= l n O n

m-1 2 n

(

. 1

(

. d[xl (s)

= x(l ) + E E_ X W ( t ) J^ X W ( s ) - j ^ ds ,

n=0 k= l n O n

and by Remark 3 we have

d [ x ] ( s )

(2.7) x (s)

ds

for se [0,1] except for s = - ~ , k=0,l,...,2 .

2m 1

Since for s-almost all x in C[0,1] and all xeD[0,l] , J v(t)5x(t) exists with

0

respect to the orthogonal development in Haar functions, then by the definition of the

P.W.Z. integral,

1

F(x) ^ J v(t)dx(t)

0

1

? lim J v (t)dx(t).

m-» 0

Thus by equation (2.6),(2.7), and (2.2) we have