A normalizing flow and its base density

Let $d\ge 1$, let $U,V\subseteq {\mathbb{R}}^d$ be nonempty open sets, and let $T:U\to V$ be a bijection such that both $T$ and $T^{-1}$ are continuously differentiable. Here $C^1$ means totally differentiable at every point with continuous derivative matrix. Such a map is a global $C^1$ diffeomorphism. A finite composition $T=T_K\circ \cdots \circ T_1$ of global $C^1$ diffeomorphisms between open subsets of ${\mathbb{R}}^d$ is a normalizing flow.

A base density on $U$ is a Lebesgue-measurable function $p_U:U\to [0,\mathrm{\infty }]$ with ${\int }_U{p}_{U}d{\lambda }_d=1$. It defines the probability measure ${\mu }_U(A)={\int }_A{p}_{U}d{\lambda }_d$ for Lebesgue-measurable $A\subseteq U$.

For the model-measure construction, assume the Axiom of Countable Choice. The model measure on $V$ is ${\mu }_X(B)={\mu }_U(T^{-1}(B))$ for Borel $B\subseteq V$. This value is defined because the inverse $C^1$ diffeomorphism maps Borel sets to Lebesgue-measurable sets under Countable Choice. When this measure has a Lebesgue density, that density is the model density.

Continuously differentiable maps, local inverses, and local diffeomorphisms — Total differentiability, continuous derivative matrices, and local diffeomorphisms.

The measure with density $f$ relative to $\mu$ — The measure defined by integrating a nonnegative measurable density.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) — Countable Choice is assumed for the model-measure construction.

Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable — Under Countable Choice, Borel subsets of Euclidean space are Lebesgue measurable.

A C^1 diffeomorphism maps Lebesgue measurable sets to Lebesgue measurable sets — A $C^1$ diffeomorphism sends Lebesgue-measurable sets to Lebesgue-measurable sets.

Papamakarios et al., §2.1, PDF pp. 2–3

Sheldon, COMPSCI 688 Lecture 23, PDF pp. 1–5