Measuring Image of a Set Using Jacobian Integral

Question

Assume $T:\mathbb{R}^d \to \mathbb{R}^d$ is a differentiable mapping and $E$ be a measurable set. Show that $m(T(E))=\int_E |det(DT(x))|dx$.

I am thinking I might use the following Theorem, but I don't know how

Let $U$ be an open subset of $\mathbb{R}^n$ and $\phi : U → \mathbb{R}^n$ be a bi-Lipschitz mapping. Let $f : φ(U) → R$ be measurable. Then $\int_U (f\circ \varphi)|\det D\varphi| = \int_{\varphi(U)}f$.

Answer 1

The measure of a set is the integral of constant function $1$ over that set (with respect to the measure). Using $$ \int_U (f\circ \varphi)|\det D\varphi| = \int_{\varphi(U)}f$$ with $f\equiv 1$ yields the result.