I was looking here: How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$? to find out how to evaluate $\displaystyle\int\limits_{-\infty}^{\infty} e^{-x^2} dx$, and didn't understand why $$\left(\displaystyle\int\limits_{-\infty}^{\infty} e^{-x^2} dx\right)\left(\int\limits_{-\infty}^{\infty} e^{-y^2} dy\right)=\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} e^{-x^2} e^{-y^2} dx\thinspace dy$$ I know you can use Fubini's Theorem from this: Why does $\left(\int_{-\infty}^{\infty}e^{-t^2} dt \right)^2= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy$?, but I'm still confused about how exactly you can just multiply two integrals together like that.
A detailed answer would be very nice.
Thanks!