Given the Lebesgue integral with the Lebesgue measure and the Borel-Sigma Algebra, I am supposed to figure out under which transformations $\int_{\mathbb{R}^2} f(x) dx$ the integral is invariant(notice that $f\in C_c(\mathbb{R}^2,\mathbb{R})$
Reflection, Rotation, Translation, Shearing.
The first two are clear, since they are isometries. Translation also preserves the volume(basically substitution will do it). Shearings should also be integral preserving, but I do not know how to properly point out why this is the case.
The matrix of a shear is $$\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix},$$ which has determinant one, so the standard change of variables formula should do the trick.