Lebesgue Measure under a Triangular or Symmetric Positive Definite Linear Map

In the setting of Euclidean Space and Lebesgue Measure: Standing Notation and Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation, let $d$ be a natural number with $1\le d$, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and let integrals of Borel functions $\mathbb{R}^{d}\to[0,\infty]$ with respect to $\lambda_{d}$ be those of Measure Spaces and the Lebesgue Integral: Standing Notation §integral, products in $[0,\infty]$ being read as in Measure Spaces and the Lebesgue Integral: Standing Notation §extended; a map from $\mathbb{R}^{d}$ into $\mathbb{R}^{d}$ or into $[0,\infty]$ is Borel in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. For a real $d\times d$ matrix $T$, the map $x\mapsto Tx$ is given by the matrix-vector product of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, and $\det T$ is the determinant of $T$. A real $d\times d$ matrix $T$ is lower triangular if $T_{ij}=0$ whenever $i<j$, and upper triangular if $T_{ij}=0$ whenever $j<i$; positive definiteness is that of that definition.

1. (Linear maps are Borel)¶ For every real $d\times d$ matrix $T$ the map $x\mapsto Tx$ from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$ is continuous and Borel.

2. (Triangular maps)¶ Let $T$ be a lower triangular or an upper triangular real $d\times d$ matrix with $0<T_{ii}$ for every $i\in[d]$. Then $0<\det T$, and for every Borel $f:\mathbb{R}^{d}\to[0,\infty]$

3. (Symmetric positive definite maps)¶ Let $A\in\mathcal{S}(d)$ be positive definite. Then $0<\det A$, the map $x\mapsto Ax$ is a bijection of $\mathbb{R}^{d}$ onto $\mathbb{R}^{d}$, and for every Borel $f:\mathbb{R}^{d}\to[0,\infty]$

In particular, for every $B\in\mathcal{B}(\mathbb{R}^{d})$ the set $A(B)=\{Ax:x\in B\}$ belongs to $\mathcal{B}(\mathbb{R}^{d})$ and $\lambda_{d}(A(B))=\det A\,\lambda_{d}(B)$.