TITLE: A product version of Riemann integral
QUESTION [6 upvotes]: Motivated by Riemann sum in Riemann integral and motivated by relations between infinite series and infinite products we ask:

Assume that $f:[0, 1]\to \mathbb{R}$ is  a positive function. Assume that there is  a real number $A$ with the following property: For every $\epsilon>0$  there is  a  partition $P=\{0=x_{0},x_{1},\ldots,x_{n-1},x_{n}=1\}$  of $[0,1]$  such that for every $t_{i}\in [x_{i-1}, \;\;x_{i}]$ we have $$| \prod_{i=1}^{n} (1+f(t_{i})\Delta x_{i})-A| < \epsilon $$

We put $A=\prod_{[0,\;1]} f $.

Is there any relation between this concept  and Riemann integrability? Is this  an appropriate generalization of Riemann integral?. Is the collection of all function $f$ for which this quantity exist both for $f^{+}$  and  $f^{-}$, an algebra of  functions? If yes, is it ismorphic to the algebra of Riemann integrable functions?

Finally can one express $\prod (f+g)$ and $\prod fg$  in term of $\prod f$   and  $\prod g$ ?

REPLY [3 votes]: Yes, there is. It is called product integral and it is exactly what you are asking. 
Moreover, if you are in a nonabelian Lie group, it does not reduce to the exponential of the integral!
See here.
Some book references:

For ordinary product integrals, here.
For Lie groups: it's usually called "ordered exponential", it is what gives parallel transport on fiber bundles and what gives the integral rotation given the angular velocity in 3d. All texts I know are physics texts, for example Weinberg's Quantum Field Theory (vol. I).