Mutual existence of product integrals in normed rings

Abstract:

Definitions and integrals are of the subdivision-refinement type, and functions are from $R \times R$ 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 $| \cdot |$ with respect to which N is complete and $|1| = 1$. If G is a function from $R \times R$ to N, then $G \in O{M^\ast }$ on [a, b] only if (i) $_x{\Pi ^y}(1 + G)$ exists for $a \leq x < y \leq b$ and (ii) if $\varepsilon > 0$, then there exists a subdivision D of [a, b] such that, if $\{ {x_i}\} _{i = 0}^n$ is a refinement of D and $0 \leq p < q \leq n$, then \[ \left |{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod \limits _{i = p + 1}^q {(1 + {G_i})} \right | < \varepsilon ;\] and $G \in O{M^ \circ }$ on [a, b] only if (i) $_x{\Pi ^y}(1 + G)$ exists for $a \leq x < y \leq b$ and (ii) the integral $\smallint _a^b|1 + G - \Pi (1 + G)|$ exists and is zero. Further, $G \in O{P^ \circ }$ on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if $\{ {x_i}\} _{i = 0}^n$ is a refinement of D and $0 < p \leq q \leq n$, then $|\Pi _{i = p}^q(1 + {G_i})| < B$. If F and G are functions from $R \times R$ to N, $F \in O{P^ \circ }$ on [a, b], each of ${\lim _{x,y \to {p^ + }}}F(x,y)$ and ${\lim _{x,y \to {p^ - }}}F(x,y)$ exists and is 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 ${\lim _{x \to {p^ - }}}G(x,p)$ exists for $p \in [a,b]$, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) $F + G \in OM^\ast$ on [a, b], (2) $F \in OM^\ast$ on [a, b], and (3) $G \in OM^\ast$ on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1) $F + G \in OM^\circ$ on [a, b], (2) $F \in OM^\circ$ on [a, b], and (3) $G \in OM^\circ$ on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].