Concerning product integrals and exponentials

Abstract:

Suppose $S$ is a linearly ordered set, $N$ is the set of real numbers, $G$ is a function from $S \times S$ to $N$, and all integrals are of the subdivision-refinement type. We show that if $\int _a^b {{G^2} = 0}$ and either integral exists, then the other exists and $a\prod {^b(1 + G) = \exp \int _a^b G }$. We also show that the bounded variation of $G$ is neither necessary nor sufficient for $\int _a^b {{G^2}}$ to be zero.