Q.1) If one wants to verify the condition that $f(t):=\frac{1}{\sqrt{(2\pi)}}exp(-t^2/2)$ is a valid pdf, one requires to verify that it's integral over the real line is 1.
While deriving that result, one method is to assume $$I = \int_{-\infty}^{\infty}f(t)dt$$
Then:
$$I^2 = \int_{-\infty}^{\infty}f(s)ds\int_{-\infty}^{\infty}f(t)dt$$ $$I^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{{(2\pi)}}exp((-t^2 - s^2)/2)dtds$$
Then use polar coordinates and show it is 1.
However, in the steps above, one uses Fubini's theorem, which requires the integrals to be finite. Hence we must first show that $I$ is indeed finite, then do the entire derivation. How does one show that $I$ is finite? Or am I wrong somewhere?
Q.2) How does one prove that the mean of the Gaussian random variable exists, i.e. $E[|X|]$ is finite, before actually computing the mean? I realize that the integral will involve an odd function, so it will be zero, but is there any other way to show finiteness, say for example by bounding it with some functions which are integrable over the real line?