Existence theorems for sum and product integrals

Abstract:

Necessary and sufficient conditions on a function G are determined for the integrals \[ \int _a^b {HG,\quad \prod \limits _a^b {(1 + HG),\quad \int _a^b {|HG - \int {HG| = 0} } } } \] and \[ \int _a^b {|1 + HG - \prod {(1 + HG)| = 0} } \] to exist, where H and G are functions from $R \times R$ to R and H is restricted by one or more of the limits $H({p^ - },p),H({p^ - },{p^ - }),H(p,{p^ + })$ and $H({p^ + },{p^ + })$. Furthermore, the conditions on G are sufficient for the existence of these integrals when H and G have their range in a normed complete ring N.