The Gaussian Weight Defines a Probability Distribution

Let $g(x)=\exp(-x^{2}/2)$ with the exponential function, let $\lambda$ be Lebesgue measure, and for a Borel set $B$ let

the integral of Lebesgue Integral of a Nonnegative Measurable Function of the measurable function $\mathbf{1}_B\,g$. Then:

1. $\nu$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (countable additivity follows from Monotone Convergence Theorem applied to the partial sums of indicators);

2. the total mass $c=\nu(\mathbb{R})=\int_{\mathbb{R}}g\,d\lambda$ is a finite positive real number; in particular the normalization $N=\nu/c$ of Standard Normal Distribution is a probability measure;

3. the function $t\mapsto\nu\bigl((-\infty,t]\bigr)$ is continuous at every point of $\mathbb{R}$, as a map from the real line $(\mathbb{R},d_{\mathbb{R}})$ into itself (single points have $\nu$-mass $0$), so the cumulative distribution function $\Phi$ of the standard normal distribution is continuous on all of $\mathbb{R}$;

4. with $\pi$ defined as the value $(\lambda\otimes\lambda)(D)$ of the product measure on the closed unit disk $D=\{(x,y)\in\mathbb{R}^2:x^{2}+y^{2}\le 1\}$ (a Borel subset of the plane for the product $\sigma$-algebra), the total mass satisfies

by Tonelli and Fubini Theorems applied to $(x,y)\mapsto g(x)g(y)$.