Since it is possible for
⋅aΠb(1 + G) to exist and not be zero when G is unbounded and 1 + G. is not
bounded away from zero, the conditions under which products of the form
|Π1n[1 + G(xq−1,xq)]| are bounded or bounded away from zero for suitable
subdivisions {xq}0n of [a,b] are important in many theorems concerning product
integrals. Conditions are obtained for such bounds to exist for products of the form
Π(1 + FG) and Π(1 + F + G), where F and G are functions from R × R to R.
Further, these results are used to obtain an existence theorem for product
integrals.
