i = k 1, j 2 1, we have Imf(R ,) I = If(0,t ) f(0,t) I. For
j = 1 1, i 2 k we have m f(Ri ) = (f(si+1,0) f(si,0)(, all as
before. Putting everything together, we have
m-1
SImf(R ,j) = If(0,0)I + EZ f(s +1,0) f(s ,0)
i,j i=k
n-1
+ E f(O,t ) f(0,tj) + E Ri (f)|
j=-1 ikk i,j
jl1
S|f(0,0)| + Var [,]f(.,0) + Var[,t]f(O,')
+ Var[(oo),(s', (f)](
= IfI(s',t') = m f (R).
Taking supremum again, we obtain Imf(R) 5S ma f(R). The proof the
other direction is similar to the ones before: for )>0, we choose a
common o,T so that
ZEf(si+1,0) f(si,0)I > Var 0,sf(.,0) -
a
EIf(O,tj+) f(O,t )I > Var tf(0,.) an
1'],t' 3 and
E 6R (f)l > Var (f) (
oT AR ,j [(0,0),(s',t')] 3
We extend and arbitrarily to partitions ',T of s,s,j
We extend a and arbitrarily to partitions oa,T' of [s,s'%