The following is a result from A Course in Abstract Harmonic Analysis by G.B. Folland. It also appears as an exercise in the real analysis text by the same author. The context is Haar measure on a locally compact group $G$.
2.21 Proposition. Suppose the underlying manifold of $G$ is an open subset of $\mathbb{R}^N$ and left translations are given by affine maps: $xy = A(x)y + b(x)$, where $A(x)$ is a linear transformation of $\mathbb{R}^N$ and $b(x)\in\mathbb{R}^N$. Then $|\det A(x)|^{-1}dx$ is a left Haar measure on $G$, where $dx$ denotes Lebesgue measure on $\mathbb{R}^N$. (Similarly for right translations and right Haar measure.)
There are several questions on this site about this result, e.g., 1, 2. However, I'm asking a different question:
Why is $|\det A(x)|^{-1}$ (as a function of $x\in G$) measurable/integrable with respect to the Lebesgue measure?
This is not included in the assumption (though in practice it is usually true) and I can't prove this.