A SIMPLE DEFINITION OF THE FEYNMAN INTEGRAL* WITH APPLICATIONS 7

b

J h(t)df(t)=0.

a

b b

For by integration by parts J h(t)df(t) = - J f(t)dh(t)=0 , since f is zero whenever

a a

h has a jump.

REMARK 2: Let h(t) be a step-function on [a,b] and let f(t) and g(t) be

continuous on [a,b] . Let f(t)=g(t) when t = a and when t=b and whenever h(t)

has a discontinuity at t . Then

b b

J h(t)df(t) = J h(t)dg(t) .

a a

REMARK 3: Let h(t) be a step-function on [0,1] whose discontinuities occur only

at the points o, -A- , -1-,-i- ,...,£- . Then

1 m-1 2n „ , 1

(2.U) h(t) = I h(s)ds + £ £ X W (t) J XW(s)h(s)ds

0 n=0 k=l n O n

for all t on [0,1] except at the points t = 0 ,

0 n= 0 k= l n O n

2m 2m '•••' 2m *

This can be easily seen because the Haar functions are a C.O.N, set. Consequently

if all the terms of the orthogonal development of h are included, it converges in the

L? mean to h . But all the non-vanishing terms in the development are included in the

(k)

right member of (2A),since h is orthogonal to X when nm . Then (2.U) holds

n

for almost all t in [a,b] , and since both members of (2.U) are continuous except at

, k=0,l,...,2 , it follows that (2.U) holds on [0,1] except at these points.

LEMMA. 2.3. Let veL^a,!)] . If

b

(2.5) F(x) = / v(t)d x(t) ,

a

then for s-almost every xec[a,b] and every xeD[a,b] , F(x) is continuous with

respect to binary polygonal approximation (continuous B.P.A.).

PROOF Case I. Let a = 0 and b = l and assume that the P.W.Z. integral is given in

terms of the Haar Functions

Let

1 m-1 2n ,,

N

1

v(t) ^ J

v

(s)ds + E Z X(k)(t) J X(k)(s)v(s)ds ,

0 n=0 k=l n O n

m-1 2 /, . 1 , .

xm(t) = x(l) + E S X

W

(t) I

XW(s)dx(s)

,

then