The covariance of the linearization is $A\mathrm{\Sigma }{A}^{\mathsf{T}}$

Given: Objects and hypotheses as in the Statement, with $A=(a_{ij})$ the Jacobian matrix of $h$ at $\mu$, so $(Av{)}_i={\sum }_{j=1}^m{a}_{ij}{v}_j$ for $v\in {\mathbb{R}}^m$.

The coordinates $X_j$ satisfy $E[X_j^2]<+\mathrm{\infty }$, the differences $X_j-{\mu }_j$ are square-integrable and integrable, and ${\mathrm{\Sigma }}_{kl}=\mathrm{Cov}(X_k,X_l)=E[(X_k-{\mu }_k)(X_l-{\mu }_l)]$ is a finite real number. (Mean vector and covariance matrix of a square-integrable random vector)

For integrable random variables $U,V$ and scalars $a,b$, $E[aU+bV]=aE[U]+bE[V]$; if $U\le V$ almost surely then $E[U]\le E[V]$; and $\mid E[U]\mid \le E[\mid U\mid ]$. In particular $\mathrm{Cov}(U,V)=E[UV]-E[U]E[V]$ for square-integrable $U,V$. (Linearity, monotonicity, and the modulus bound for expectation, Variance and covariance identities for random variables)

Covariance is symmetric and bilinear on finite linear combinations of square-integrable real random variables: for finite families $(U_i)$, $(V_j)$ and real scalars, $\mathrm{Cov}({\sum }_i{a}_i{U}_i,{\sum }_j{b}_j{V}_j)={\sum }_{i,j}{a}_i{b}_j\mathrm{Cov}(U_i,V_j)$, and covariance with a constant is zero. (Variance and covariance identities for random variables)

Each coordinate projection on $U\subseteq {\mathbb{R}}^m$ is continuous, hence has Borel preimages of Borel sets; finite sums and scalar multiples of measurable real-valued functions are measurable, and a map into ${\mathbb{R}}^d$ is measurable exactly when its coordinates are. (A continuous map has Borel preimages of Borel sets, Closure properties of measurable functions used by the integral, A map into ${\mathbb{R}}^n$ is measurable exactly when its coordinates are measurable)

$(Av{)}_i={\sum }_{j=1}^m{a}_{ij}{v}_j$ and $(A\mathrm{\Sigma }{A}^{\mathsf{T}}{)}_{ij}={\sum }_{k,l=1}^m{a}_{ik}{a}_{jl}{\mathrm{\Sigma }}_{kl}$ in the matrix space. (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication)

For vectors $u,v$ in a real inner-product space, $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \mathrm{\parallel }u\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$. (Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs)