A product version of Riemann integral

Question

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 $\prod g$ ?

Answer 1

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).