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
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