- (f(s'+,t'+) f(h+,t'+) f(s'+,k+) + f(h+,k+)I
< f(s'+,t'+) f(s'+,t'+)I + If(h+,t'+) f(s+,t'+)
+ |f(s'+,k+) f(s'+,t+)l + If(h+,k+) f(s+,t+)|
C E E E
hence o is inner regular on 6. It follows, then, by Proposition 19
[6, p. 314], that JoI is also inner regular on 6.
5) Let r(6) be the class of subsets MC R2 for which M J E 6
for any J e 6. Since a is additive on 6, Jla is additive on t(6)
(standard result from measure theory), hence Jol is additive
on 6C (6). We shall now denote Jo| by V (for clarity in what
follows).
Let o, V be the additive set functions obtained (uniquely) by
extending a and V to the ring C generated by 6. We show next that
n
o has finite variation on C. In fact, let A E C. Then A = ) A ,
1=1
Ai disjoint, Ai E 6. We have
n n
lo(A)l lo( A )I = | I o(Ai ) 5 E lo(A1 ) =
i i=1 i-1
n n n n
= E o(Ai) ) I V(A ) = E V(Ai) = V( Ai) = V(A).
1-1 i=1 i=1 i=1
Then lo(A)I V(A) for all A c C, so lal S V (since Joi is the
smallest positive measure dominating lo(.) ), hence o has finite
variation.