 
Summary: 1. Integration.
Definition 1.1. We say f : R2
R is admissible if f is bounded and {(x, y)
R2
: f(x, y) = 0} is bounded.
Definition 1.2. Suppose f : R2
R is admissible and 0 < < . We let
R(f)
be the set of sums
S =
M
i=1
N
j=1
f(i, j)(xi  xi1)(yj  yj1)
where
(i) M and N are positive integers;
(ii) x0 x1 · · · xM and y0 y1 · · · yN ;
(iii) xi  xi1 for i = 1, . . . , M and yj  yj1 for j = 1, . . . , N;
(iv) xi1 i xi for i = 1, . . . , M and yj1 j yj for j = 1, . . . , N;
