I am searching for something of a probability analogue to integration:
$$\prod_{x_0}^{x_1} p(x)^{dx}$$
( Using product sign in lack of anything else. )
I suppose it would make sense to define it to be the exponent of the following:
$$\int_{x_0}^{x_1} \log( p(x))dx$$
Using the fact how the logarithm laws behave with the discrete counterparts:
$$\exp \left( {1\over n} \sum_{k=1}^n\log(p(k))\right) = \sqrt[n]{\prod_{k=1}^{n} p(k)}$$
But I am unsure if it would be easy or hard to make strictly mathematical correct. What do you think, is it possible to make formal. If it is, what sense would it make?