An existence theorem for sum and product integrals

Abstract:

Functions are from $S \times S$ to $N$, where $S$ and $N$ denote a linearly ordered set and a normed ring, respectively. Theorem. If $\{ a,b\} \in S \times S,G$ has bounded variation on $\{ a,b\}$, $\{ {F_n}\} _1^\infty$ converges uniformly to a bounded function $F$ on $\{ a,b\}$ and either \[ (1){\quad _x}{\prod ^y}(1 + {F_n}G)\quad and\quad \int _a^b {|1 + {F_n}G - \prod {(1 + {F_n}G)} |} \] exist for $n = 1,2, \cdots$ and each subdivision $\{ a,x,y,b\}$ of $\{ a,b\}$ and $\{ \smallint _a^b|1 + {F_n}G - \prod ( 1 + {F_n}G)|\} _1^\infty$ converges to zero or \[ (2)\quad \int _a^b {{F_n}G\quad and \quad \int _a^b {|{F_n}G - \int {{F_n}G|} } } \] exist for $n = 1,2, \cdots$ and $\{ \smallint _a^b|{F_n}G - \smallint {F_n}G|\} _1^\infty$ converges to zero, then (conclusion) $_a\prod {^b} (1 + FG){\;_a}\prod {^b} (1 + |FG|),\smallint _a^bFG$ and $\smallint _a^b|FG|$ exist. Further, $\smallint _a^b|1 + FG - \prod {(1 + FG)|,\smallint _a^b|1 + |FG| - \prod {(1 + |FG|)|,\smallint _a^b|FG - \smallint FG|} }$ and $\smallint _a^b||FG| - \smallint |FG||$ exist and are zero.