Standard normal and normal laws

Assume AC. Define $\gamma (E)={\int }_E{e}^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }dx$ for Borel E in $\mathbb{R}$. By The standard normal density has total mass one and The indefinite integral of a nonnegative measurable function is a measure, gamma is a probability measure; denote it $N(0,1)$. For $m\in \mathbb{R}$ and $\sigma \ge 0$, define $N(m,{\sigma }^2)$ as the law of $x\mapsto m+\sigma x$ on $(\mathbb{R},\mathcal{B},\gamma )$. This affine map is continuous: for $\sigma$>0 choose $\delta$=$\epsilon$/$\sigma$, and for $\sigma$=0 it is constant. Its inverse images of opens are open, so it is Borel measurable. The law of a random element is a probability measure makes its pushforward a probability. When $\sigma$=0, the preimage of E is all of R if m belongs to E and empty otherwise, so $N(m,0)={\delta }_m$ in The Dirac set function at a point.

The standard normal density has total mass one

The indefinite integral of a nonnegative measurable function is a measure

The law of a random element is a probability measure

The Dirac set function at a point