rectangle R,1 = [(st'),(sCs + ,tl+1)] Q', and we have
IAR f IA fil |A If AR f
k,l k,1 k, k,l
Sf(sk+l tl+ )-f(sk+1 ,t)-f(sk 1 )+f(s t)-f(s+1 ,t +
+ (s + ,t;)+f(s',t + )-f(s ,t')!
If(sk+1't l+1 )-f(sk I't+l)l + If(Sk+1 tl)-f( +l1't )
+ If(skt +1 )-f(s ,t+1)1 + If(sktl)-f(sktl)
C + + E:n+ C =CE:
4mn 4mn 4mn Nmn mn
Summing up, then, we obtain
E |& fI E IAR fl = I | (AR f| |AR fI
R EQ 1 R .Q' O O, there is a grid QN with
Var z,z](f;Q ) > N + 1. By the above, there exists a grid QN with
rational coordinates such that Var [zz'(f;QN) > N + > N,* hence
Lz,zJ N 2
sup Var ,z(f;Q) =+ Var[ ,z(f). Similarly, if
Q=oxT zz] zz
Q rational
Var[zz ](f) = a < m, for every e > 0 there exists Q = oxT such that
Var ,(f;Q ) > a and there exists Q' = o'xT' with rational
[z,z 'I E 2 E
* Again, Q' is not a partition of R, so this must be interpreted
directly as the sum given in Definition 2.2.4(a).