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I am afraid, that multiplication of even countably many small numbers yields 0. Let alone the product of more than that, what your integration analogous operation would be,
You can get a nonzero product if the sum of differences between 1 and your factors converge. Then and only then. But if all the factors are smaller than say 0.9 ... you get 0.
Except if you can find some creative way to that anyway. Might be possible, I don't know.
Yeah, it might have helped to clarify that the infinitesimal factors I had in mind are not infinitely small as numbers from the standpoint of addition. Since the factor that makes no change to the product is 1 rather than 0, "infinitely small" factors must be infinitesimally greater than 1, not 0. In particular, I was talking about a Type II product integral with the formula pi(1 + f(x).dx). If f(x) = 1, then we get e^sigma(1.dx) = e^constant = constant, right?