I am trying to understand the justification of step (*) below, which I have seen used to find the integral of the Error Function Er(f); the cdf of the standard-normal normal .
Here is the derivation I know:
We set $I:= \int_{-\infty}^\infty e^{-x^2}dx$
Then we use: $I^2= \left( \int_{-\infty}^\infty e^{-x^2} dx\right) \left(\int_{-\infty}^\infty e^{-y^2} dy\right)$
Then we end up with: (*) $I^2=\int_{-\infty}^\infty e^{-(x^2+y^2)} dxdy$ --and we have a nice polar integral. question:
How do we conclude that $\big(\int f(x)dx\big) \big(\int f(y)dy\big) = \int f(x)f(y) dxdy$?
AFAIK , integration is not multiplicative. What gives?