Is there any notion of taking a continuous product of the values of a real function over an interval? Only thing I could think of was exponentiating to the power of the integral over that interval.
If in some way we have $\sum a_n\to\int a_n ~ dx$ for a suitable definition of "$\to$"
Then $\prod a_n = \exp\left(\log \prod a_n \right)=\exp\left(\sum \log(a_n)\right)\to\exp\left(\int \log(a_n)~dx\right)$
Edit: after googling I found this
which might interest you