and [T,T'], respectively. The same computation as before gives
E jmf(Rj ) = ff(0,0)I + Zlf(si+1,0) f(si,0)D + ZBf(0,t )
i,j 0 T
f(o,t j) + Z |AR (f)|
oxT i,j
> jf(0,0)l + Var[0,s f(.,0) + Var[Ot']
+ Var (f)
[(0,0),(s',t')] 3
Ifl(s',t') E = m lf(R) E.
Hence Imf(R)I > mlfl(R) e. Letting e 0, we obtain
Im (R)l a m f(R); hence equality.
This takes care of all the possibilities, so we have Imfl = mlf
of S. Moreover, both are o-additive on S; the first since mf is by
Theorem 3.1.1, the second since Ijf is right continuous by Theorem
2.3.2. Since Imfl, mlf are equal and o-additive on S, they are equal
on a(S) B(R2), and the theorem is proved. x