Existence of sum and product integrals

Abstract:

Functions are from $R \times R$ to R, where R represents the set of real numbers. If c is a number and either (1) $\smallint _a^b{G^2}$ exists and $\smallint _a^bG$ exists, (2) $\smallint _a^bG$ exists and $_a{{\mathbf {\Pi }}^b}(1 + G)$ exists and is not zero or (3) each of $_a{{\mathbf {\Pi }}^b}(1 + G)$ and $_a{\Pi ^b}(1 - G)$ exists and is not zero, then $\smallint _a^bcG$ exists, $\smallint _a^b|cG - \smallint cG| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + cG)$ exists for $a \leq x < y \leq b$ and $\smallint _a^b|1 + cG - {\mathbf {\Pi }}(1 + cG)| = 0$. Furthermore, if H is a function such that ${\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)$ and ${\lim _{x,y \to {p^ + }}}H(x,y)$ exist for each $p \in [a,b],n \geq 2$ is an integer, and G satisfies either (1), (2) or (3) of the above, then $\smallint _a^bH{G^n}$ exists, $\smallint _a^b|H{G^n} - \smallint H{G^n}| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + H{G^n})$ exists for $a \leq x < y \leq b$ and $\smallint _a^b|1 + H{G^n} - {\mathbf {\Pi }}(1 + H{G^n})| = 0$.