Positive-definite multivariate Gaussian density is normalized

Set $A:={\mathrm{\Sigma }}^{1\mathrm{/}2}$, the symmetric positive-definite square root in the Statement, so $A^2=\mathrm{\Sigma }$. By [F8], write $A=U\mathrm{diag}(a_1,\dots ,a_p)U^{\mathsf{T}}$ with $U^{\mathsf{T}}U=I$ and $a_j>0$. The determinant laws [F11], including the triangular formula for the diagonal matrix, give $(\det U{)}^2=\det (U^{\mathsf{T}}U)=\det I=1$ and $\det A=(\det U{)}^2{\prod }_j{a}_j={\prod }_j{a}_j>0$. Also $\det \mathrm{\Sigma }=\det (A^2)=(\det A{)}^2$, so $\det A=\sqrt{\det \mathrm{\Sigma }}$. In the same basis, ${\mathrm{\Sigma }}^{-1}=A^{-2}=U\mathrm{diag}(a_1^{-2},\dots ,a_p^{-2})U^{\mathsf{T}}$ and $A^{\mathsf{T}}{\mathrm{\Sigma }}^{-1}A=I$.

By [F2] and [F4], the joint law of $Z$ is the product of its standard-normal marginals. Define $g(y):={\prod }_{j=1}^p{\varphi }_1(y_j)=(2\pi {)}^{-p\mathrm{/}2}{e}^{-\mathrm{\parallel }y{\mathrm{\parallel }}_2^2\mathrm{/}2}$. Tonelli [F5] and the product-measure identity [F6] give ${\int }_{{\mathbb{R}}^p}gd{\lambda }_p={\prod }_{j=1}^p{\int }_{\mathbb{R}}{\varphi }_{1}d{\lambda }_1=1$ and, for every Borel rectangle $R={\prod }_{j=1}^p{B}_j$, ${\int }_{R}gd{\lambda }_p={\prod }_{j=1}^p{\int }_{B_j}{\varphi }_{1}d{\lambda }_1$. Thus $gd{\lambda }_p$ and the product law of $Z$ are probability measures agreeing on rectangles. The rational half-open rectangles form a generating pi-system; $(-N,N{]}^p$ is an increasing exhaustion from it, and the common rectangle values make their measures equal and finite on every such box. Hence [F7] identifies the measures. AC supplies countable choice for the product-Lebesgue identity [F6].

Let $T(y)=\mu +Ay$. By step 1.1, $A$ is invertible, so $T$ is a $C^1$ diffeomorphism of ${\mathbb{R}}^p$ with $\mathrm{\mid }\det DT\mathrm{\mid }=\det A$. The linear measure-scaling result [F10] confirms this is the Lebesgue volume factor. The displayed function ${\varphi }_p$ is continuous and positive, so $f(x)={\mathbf{1}}_E(x){\varphi }_p(x;\mu ,\mathrm{\Sigma })$ is nonnegative Borel for every Borel $E\subseteq {\mathbb{R}}^p$. AC supplies the countable-choice assumption of [F9] by [F12]. Apply [F9] to $f$ and the change of variables by $T$ to obtain ${\int }_E{\varphi }_p(x;\mu ,\mathrm{\Sigma })dx={\int }_{T^{-1}(E)}{\varphi }_p(T(y);\mu ,\mathrm{\Sigma })\det Ady\mathrm{.}$ For $x=T(y)$, $x-\mu =Ay$ and $A^{\mathsf{T}}{\mathrm{\Sigma }}^{-1}A=A{A}^{-2}A=I$. Thus the exponent in [F1] becomes $-\mathrm{\parallel }y{\mathrm{\parallel }}_2^2\mathrm{/}2$, while $(\det \mathrm{\Sigma }{)}^{-1\mathrm{/}2}\det A=1$ by step 1.1. Hence ${\varphi }_p(T(y);\mu ,\mathrm{\Sigma })\det A=g(y)$. The last integral equals ${\int }_{T^{-1}(E)}g(y)dy=\mathbb{P}(X\in E)$ by step 1.2 and the definition of $X$. Thus ${\varphi }_p$ is the Lebesgue density of $X$.

By [F3], $X$ has law $N_p(\mu ,\mathrm{\Sigma })$. Step 2.1 identifies that law on every Borel set with integration against ${\varphi }_p$; taking $E={\mathbb{R}}^p$ gives ${\int }_{{\mathbb{R}}^p}{\varphi }_p(x;\mu ,\mathrm{\Sigma })dx=\mathbb{P}(X\in {\mathbb{R}}^p)=1$. Nonnegativity follows directly from [F1] and $\det \mathrm{\Sigma }>0$ in step 1.1.