Let $g(x)=\exp(-x^{2}/2)$ with the \reftext{def:exponential-function-real-2026a}{exponential function}, let $\lambda$ be \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure}, and for a \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} $B$ let

the integral of \ref{def:lebesgue-integral-nonnegative-2026a} of the \reftext{def:measurable-function-2026a}{measurable} function $\mathbf{1}_B\,g$. Then:

1. $\nu$ is a \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (countable additivity follows from \ref{thm:monotone-convergence-2026a} 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 \ref{def:standard-normal-distribution-2026a} is a probability measure;

3. the function $t\mapsto\nu\bigl((-\infty,t]\bigr)$ is \reftext{def:continuous-at-point-c54-2026b}{continuous} at every point of $\mathbb{R}$ (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 \reftext{thm:product-measure-2026a}{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 \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra}), the total mass satisfies

by \ref{thm:tonelli-fubini-2026a} applied to $(x,y)\mapsto g(x)g(y)$.