Weak convergence of gaussian laws by parameters

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 lem-normal-density-has-total-mass-one and thm-indefinite-integral-of-a-nonnegative-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. lem-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 def-dirac-measure.

Dominated convergence: Let $f$ and $(f_n)$ be measurable complex-valued functions such that $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single nonnegative measurable function $g$ with $\int gd\mu <+\mathrm{\infty }$. Then $f\in L^1(\mu )$, $\int \mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu ⟶0,$ and hence $\int f_{n}d\mu ⟶\int fd\mu \mathrm{.}$

Change of variables for expectation: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element, let ${\mathbb{P}}_X$ be its law, and let $g:(S,\mathrm{\Sigma })\to \mathbb{R}$ or $g:(S,\mathrm{\Sigma })\to \mathbb{C}$ be measurable.

1. If $g\ge 0$, then $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$
2. If $g(X)$ is integrable, then $g$ is integrable with respect to ${\mathbb{P}}_X$ and the same formula holds: $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Weak convergence of borel probability measures: For Borel probability measures ${\mu }_n,\mu$ on a metric space S, write ${\mu }_n\Rightarrow \mu$ if $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function f on S. Continuity is def-metric-continuity. Such f is Borel measurable (inverse images of open sets are open) and $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mu (S)<\mathrm{\infty }$, so the integrals are finite in def-integrable-real-and-complex-functions-and-their-integrals. No completeness or coupling is required.