Gaussian kernels form an approximate identity

Given: Countable Choice, $n\ge 1$, and $\delta >0$ wherever it appears.

Countable Choice is the hypothesis carried by the cited integration interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

For every $t>0$ the kernel satisfies $\mathrm{\Gamma }(\cdot ,t)>0$, ${\int }_{{\mathbb{R}}^n}\mathrm{\Gamma }(x,t)dx=1$, and the parabolic scaling identity $\mathrm{\Gamma }(\lambda x,{\lambda }^{2}t)={\lambda }^{-n}\mathrm{\Gamma }(x,t)$ for every $\lambda >0$ and $x\in {\mathbb{R}}^n$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

An $L^1$ approximate identity on ${\mathbb{R}}^n$ is a family $(K_{\epsilon }{)}_{\epsilon >0}\subseteq L^1({\mathbb{R}}^n)$ with $\int K_{\epsilon }=1$, with $\mathrm{\parallel }{K}_{\epsilon }{\mathrm{\parallel }}_1$ bounded independently of $\epsilon$, and with ${\int }_{\mathrm{\mid }x\mathrm{\mid }>\delta }\mathrm{\mid }{K}_{\epsilon }\mathrm{\mid }\to 0$ as $\epsilon \to 0^{+}$ for every $\delta >0$ (An $L^1$ approximate identity on ${\mathbb{R}}^n$).

If $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single integrable nonnegative $g$, then $\int f_n\to \int f$ (Dominated convergence).

For a $C^1$ diffeomorphism $T:U\to V$ of open sets and every nonnegative Lebesgue measurable $f:V\to [0,\mathrm{\infty }]$, ${\int }_{V}f(y)dy={\int }_{U}f(T(x))\mathrm{\mid }\det DT(x)\mathrm{\mid }dx$; the scaling $x=\sqrt{t}z$ is such a diffeomorphism with $\mathrm{\mid }\det DT\mathrm{\mid }=t^{n\mathrm{/}2}$ (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, A linear map $T$ of ${\mathbb{R}}^n$ sends Lebesgue measurable sets to Lebesgue measurable sets, with ${\lambda }_n(T[E])=\mathrm{\mid }\det T\mathrm{\mid }{\lambda }_n(E)$ when $T$ is invertible and $T[E]$ Lebesgue null when it is not).