+ [Var [0,]f(0,*) Var[0,t]f(0)] = Var[(0,t),(st)]
+ Var t,f(O,-). Taking supremum over grills oxT on both sides, we
get (as before) Imf(R)I 5 mfI (R).
On the other hand, for any E>0, there exists a partition T' of
n-1
[t,t'] such that E If(f((,t O,t)I > Var f(0,) and
J- Q J+1 J Lt,t J 2
j-0
a grill o0xr0 of [0,s']x[t,t'] such that ZIAR (f) >
1,j
Var [,t), .)(f) We choose a common refinement T of T
[(0,t),(s',t')] 2
and T., and extend 00 arbitrari:.y to get a partition o of [s,s'].
Then, for the grill oxT, we have
n-1
E Imf(Rj .)I = ZE f(0,tj+,) f(0,t ) + E A (f) I (as before)
l,j j=0 a XT i,j
> Var[t, f(O, + Var[(,), (f) -
[t,tfl 2 [(OIt)I(s',2
= m fl(R) E.
Since the left hand side is bourded above by Imfl(R) for any grill,
we have Imfl(R) > mlfl(R) c. Letting e 0 again, we get
Im f(R) mi (R), hence equality.
The next case proceeds similarly.
3) (s,t) in the fourth quEdrant: sa0, t<0. This time, we
refine T by including zero if necessary, and compute variations along
such grills. Denoting tk = 0 a. before, we have (same computation as
before):