Exercise 1.3.16 (Linear change of variables in Lebesgue integral)

Question

Let $f:\mathbb{R}^d	o[0,+\infty]$ be a Lebesgue measurable function, and let $T:\mathbb{R}^d	o\mathbb{R}^d$ be an invertible linear transformation. Show that
$$\int_{\mathbb{R}^d} f(T) = \frac{1}{|\det T|} \int_{\mathbb{R}^d} f$$
or equivalently $\int_{\mathbb{R}^d} f(T^{-1}) = {|\det T|} \int_{\mathbb{R}^d} f$.

Elu Thingol · Accepted Answer

To demonstrate that
$$\int_{\mathbb{R}^d} f(T) = \frac{1}{|\det T|} \int_{\mathbb{R}^d} f $$
means to show that
$$|\det(T)|	imes \inf \left\{ \mathrm{Simp}\int_{\mathbb{R}^{d}} g: \begin{array}{l} g 	ext{ is simple}\ f\circ T \leq g \end{array} ight\} = 
\inf \left\{ \mathrm{Simp}\int_{\mathbb{R}^{d}} h: \begin{array}{l} h 	ext{ is simple}\ f \leq h \end{array} ight\}$$

NOT FINISHED.

Exercise 1.3.16 (Linear change of variables in Lebesgue integral)

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Exercise 1.3.16 (Linear change of variables in Lebesgue integral)

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