Mutual existence of product integrals

Abstract:

Definitions and integrals are of the subdivision-refinement type, and functions are from $R \times R$ to R, where R represents the real numbers. Let $O{M^ \circ }$ be the class of functions G such that $_x\prod {^y} (1 + G)$ exists for $a \leqq x < y \leqq b$ and $\smallint _a^b|1 + G - \prod {(1 + G)| = 0}$. Let $O{P^ \circ }$ be the class of functions G such that $|\prod \nolimits _{q = i}^j {(1 + {G_q})|}$ is bounded for refinements $\{ {x_q}\} _{q = 0}^n$ of a suitable subdivision of [a, b]. If F and G are functions from $R \times R$ to R such that $F \in O{P^ \circ }$ on [a, b], ${\lim _{x,y \to {p^ + }}}F(x,y)$ and ${\lim _{x,y \to {p^ - }}}F(x,y)$ exist and are zero for $p \in [a,b]$, each of ${\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)$ and ${\lim _{x \to {p^ - }}}G(x,p)$ exist for $p \in [a,b]$, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) $F + G \in O{M^ \circ }$ on [a, b], (2) $F \in O{M^ \circ }$ on [a, b], and (3) $G \in O{M^\circ }$ on [a, b].