For all real \(x,y\), let \(\varphi (t)=(2\pi )^{-1/2}e^{-t^2/2}\) and \(d_2(x,y)=(2\pi )^{-1}e^{-(x^2+y^2)/2}\). Then \(\varphi (x)\varphi (y)=d_2(x,y)\), with both sides regarded as nonnegative extended real values.

Let \(\gamma \) be the standard real Gaussian probability measure. For every measurable \(A\subseteq \mathbb R^2\), \((\gamma \otimes \gamma )(A)=\int _A(2\pi )^{-1}e^{-(x^2+y^2)/2}\, dx\, dy\), where the integral is nonnegative and extended-valued.

Let \(f(r)=r\exp (-r^2/2)\) for \(r\gt 0\). Its nonnegative extended integral satisfies \(\int _{(0,\infty )}f(r)\, dr=1\).

Let \(A\subseteq \mathbb R^2\) and \(\Theta \subseteq \mathbb R\) be measurable sets such that for all real \(r\gt 0\) and \(\theta \), \((r\cos \theta ,r\sin \theta )\in A\) if and only if \(\theta \in \Theta \). Then \(\int _A(2\pi )^{-1}e^{-(x^2+y^2)/2}\, dx\, dy=(2\pi )^{-1}\operatorname {Leb}(\Theta \cap (-\pi ,\pi ))\), with the integral and angular Lebesgue measure in the nonnegative extended reals.

For every real \(\theta \) with \(-\pi \lt \theta \lt \pi \), \(\cos \theta \le 0\) if and only if \(\theta \le -\pi /2\) or \(\pi /2\le \theta \).

For every real \(u\) with \(-2\pi \lt u\le -\pi /2\), \(\cos u\le 0\) if and only if \(-3\pi /2\le u\).

For every real \(u\) with \(-\pi /2\le u\lt \pi \), \(\cos u\le 0\) if and only if \(\pi /2\le u\) or \(u=-\pi /2\).

For every real \(\alpha \), the set \(\{ \theta \in \mathbb R:\cos \theta \le 0\ \text{and}\ \cos (\theta -\alpha )\le 0\} \) is Borel measurable.

For every real \(\alpha \in [0,\pi ]\), the Lebesgue measure of \(\{ \theta \in (-\pi ,\pi ):\cos \theta \le 0,\ \cos (\theta -\alpha )\le 0\} \) equals \(\pi -\alpha \).