Let $(\mathbb{R}^d,\mathcal{L},m)$ be the Lebesgue measure space, and let $T:\mathbb{R}^d\to\mathbb{R}^d$ be an invertible linear transformation. Show that $$T_*m = \frac{1}{\det(T)} m.$$

Homepage › Solution manuals › Terence Tao › An Introduction to Measure Theory › Exercise 1.4.38 (Linear change of variables as pushforward meusure)

Exercise 1.4.38 (Linear change of variables as pushforward meusure)

Let $(ℝ^{d}, ℒ, m)$ be the Lebesgue measure space, and let $T : ℝ^{d} \to ℝ^{d}$ be an invertible linear transformation. Show that

T_{∗} m = \frac{1}{\det (T)} m .