R if Q is a refinement of P. Prop. 2.2.3 says, then, that the class

PR of partitions of R is directed under this order.

 We are now ready to define the variation of a function on a

rectangle.

Definition 2.2.4. Let R = [z,z'] = [(s,t),(s',t')] be a closed,
 2 2
bounded rectangle in R and let f: R + E be a function.

 a) For P = (Rj)) a partition of R, R = [(s ,t ),(s ,t )], we
 ~ JE J J i 3 3
 define


 Var zz'] (f;P) = IAR fl
 JEJ J

 E (f(sj,tj)-f(s ,t )-f(s ,tj)+f(s ,t ) .
 jEJ

 b) We define the variation of f on R, denoted Var z,z'(f), by


 Var ,] (f) = sup Var (f;P) S +
 [z,z ] [z,z']
 PC
 R
Remark. The supremum in part (b) always exists (finite or infinite),

since the map P Var[z,z (f;P) is increasing for the order defined

above on P To see this, consider a rectangle [z,z'] partitioned

into two rectangles R1 and R2, as in Figure 2-1.



 t






 t
 s s5


Figure 2-1 A partition of [z,z']