A SIMPLE DEFINITION OF THE FEYNMAN INTEGRAL, WITH APPLICATIONS 9
1
F(x) = lim J v ( t ) x
n
( t ) d t
n
- o o
0
1 d[x] (t)
= lim J v(t) ~ dt
n - » 0
1
= lim J v(t)d[x]
n
(t)
n
- c o
0
= lim F( [x] ) for s-almost a l l x e c [ a , b ] and a l l x€D[a,b ] .
Case I I : Let a = 0 and b = l , and l e t the P.W.Z. integra l be defined by any C.O.N.
sequence [cp 3 of functions of bounded variation . Then by the theorem of Paley, Wiener
and Zygmund, the integra l
[cpn3 i _ {X£k)} i
J v(t)dx(t) - n J v(t)dx(t)
0 0
for almost all xec[0,l] , and since the integral is linear, this holds for s-almost all
xeC[0,l] . Moreover by (2.2) the above equation holds for all xeD[0,lj .
Case III: Let [a,b] be any interval, and the C.O.N, sequence be unrestricted. By a
translation and a change of scale from [0,1] to [a,b] the result follovs, since scale
invariant Wiener null sets go into scale invariant Wiener null sets by such transforma-
tions, and the Lemma is proved.
—» THEOREM 2.1 If
F(x)eS•x-
, then F(x) is continuous with respect to binary polygonal
v v
approximation s-almost everywhere in C [a,b] and everywhere in D [a,b].
PROOF: Since F e S , there exists * e 7* such that
(2.8) F(x) = J
v
expU Z J v (t)dx. (t)}d|j,(v)
L2 j=l a
(It is assumed that the P.W.Z. integrals in the exponential are based on the same
C.O.N, sequence ityJi of functions of bounded variation for all v.eLp and all
Xj€C[a,b] (j=l,2,...,v)) .
Then substituting [x ] for x , we have
v b
F([xl ) = J exp{i £ J v (t)d[x ] (t)}dH(v) .
m L2 j=l a ° J m
By Lemma 2.3j the above exponential approaches the exponential in (2.8) as m-*00 , so by
-• v v
bounded convergence and because the exponential is measurable in vX x on k x C , we
have F( [x] ) —F(x) for s-almost a l l x e C [a,b] and every x e D [a,b] , and the
theorem i s proved.
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