A Differentiation Theorem for Additive Processes

203

We now apply the Chacon-Ornstein filling scheme [4, Lemma 1] with -F instead of/"^ and g instead of/ 1 to obtain functions dQ,...,dj^_^ such that

'^ 1

X T^^-'dj^TJ^ ~F^ for all /c = 0,...,X-l,

and such that d-' = dQ-\------hJ|^_i=g on E'. Since de^^^if) this shows that

Ч'^' { F ) ^^dd^i^^dd^л=jgd|л^igd^i - 2£

E E E E

and completes the proof.

( 2 . 8 ) Lemma. Let F and G be two positive additive processes and assume that sup (F;-G,)>0 on a set E with ^u(£)<oo. Then for each e>0 there exists a set

teQito )

E'czE and a number ô>0 such that pl{E — E')<e and such that sup {F^-S,g)>{) on E' for all ge^^(G).

teQito )

Proof We find finitely many t^eQito), I Si un, and an a>0 such that if E^ = En{F,-G^>oc}

then / / ( £ - U£ , ) <^ .

Since G is strongly continuous and G^^^ decreases to G^ when seQ{tQ) decreases to zero there exists a ô>0 such that the sets Б^ = {С^^^^ —G^ >a} have

g measure /z(5J<-—(i = 1, ...,n). Now, if ge^^G), then, by Lemma2.6

2n

f , , - S , , gèf , - G , , , , = (F,_-G,,)-(G,,^,-G,,)>0

n

on E\B^. Therefore E'= [j (E\B^) has the desired property.

1= 1

( 2 . 9 ) Lemma. Let F and G be two positive additive processes and assume that sup (F^ —G,)>0 on E for all tQ>0. Assume that fi{E)<cc. Then, for each e>0

teQito )

there exists a set E'czE such that fi{E — E')<8 and such that 'F^ ,(F)^ ^£,,(G) whenever E"e^ and E"ciE'.

00

Proof We choose г^>0 with X ^i<^ ^^^ ^^^^ ^^^ ^^^^ ^Л^- ^^^ previous

Lemma shows that for each / we can find a set F^ciF and a number (5^>0 such that ^{E-E^)<s^ and such that sup (F,-5,g)>0 on E^ for all ge^^ (G). Let F'

= Pi F,. Then /i(F-F')<e. 1= 1 Let E" ciE' be a fixed set. Given an f/>0 we find an г such that %'{F)S%„{F) + f]. Hence, if ge^.^G) then, by Lemma (2.8), W'^)\F)^ j g dix and consequently ^^ ,(F) + ^ ^ ^/' (G). ' ^"