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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?

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    $\begingroup$ Have you seen this? en.wikipedia.org/wiki/Product_integral $\endgroup$ – Dan Brumleve Mar 8 '17 at 20:45
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    $\begingroup$ The exponential of the integral of the logarithm is indeed a sort of "multiplicative integral". One place I've seen this used is in Szego's asymptotic formula for the determinant of certain large matrices. $\endgroup$ – Ian Mar 8 '17 at 20:53

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