Define $g:\mathbb{R}\to\mathbb{R}$ by $g(x)=\exp(-x^{2}/2)$, with the \reftext{def:exponential-function-real-2026a}{exponential function}; $g$ is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous} (a composition of continuous maps, by \ref{thm:composition-continuous-euclidean-2026a} and claim 3 of \ref{thm:exponential-properties-2026a}), hence \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra} by the generator criterion there (preimages of open sets are open).

By claim 2 of \ref{thm:gaussian-integral-2026a}, the quantity

(the integral of \ref{def:lebesgue-integral-nonnegative-2026a} with respect to \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure} $\lambda$) is a finite positive real number. The \textbf{standard normal distribution} is the function

with $\mathbf{1}_B$ the indicator of \ref{def:simple-function-integral-2026a}. By claims 1 and 2 of \ref{thm:gaussian-integral-2026a}, $N$ is a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Its \reftext{def:distribution-cdf-random-variable-2026a}{cumulative distribution function} is denoted

and $\Phi$ is continuous at every point (claim 3 of \ref{thm:gaussian-integral-2026a}). A random variable is called \textbf{standard normal} (or \textbf{standard Gaussian}) if its distribution is $N$.