Solution of Integral Equations by Product Integration

Lower case letters are used to represent functions from R to N, and capital letters are used to represent functions from R x R to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm | • | with respect to which N is complete and |l| = 1. For a subdivision \x.\n. of an interval [a, b], we use G. and /. to denote G(x. x.) and f(x), respectively. The statement that G £ OB on [a, b] means there exist a subdivision D of [a, b] and a number B such that, if i*l"=0 is a refinement of D, then 1" , \G.\ < B. i=l i i> The statement that f G exists means there exists an element L oí N Ja such that, if e > 0, then there exists a subdivision D of [a, b] such that, if ,x¿S"=0 is a refinement of D, then |L 2TM=1 G¿| < e. Further, G e OA° on [zz, b] only if Q G exists and ^ \G / G| = 0. The statement that II (1 + G) exists means there exists an element L