In Exercise 1.1 of Lie Groups written by Daniel Bump, let $d_{\mathrm{a}}X$ denote the Lebesgue measure on $\mathrm{Mat}_n(\mathbb{R})$. I feel confused to understand the definition and properties of such a $d_{\mathrm{a}}X$. Questions are as follows.
- How can I understand the measurable sets on $\mathrm{Mat}_n(\mathbb{R})$ and whether it is open or compact? And how to verify that $d_{\mathrm{a}}X$ is a Haar measure for the additive group $\mathrm{Mat}_n(\mathbb{R})$?
- To show that $|\mathrm{det}(X)|^{-n}d_{\mathrm{a}}X$ is both a left and a right Haar measure on $\mathrm{GL}(n,\mathbb{R})$, should I check that $$\int_{\mathrm{Mat}_n{\mathbb{R}}}f(AX)\mathrm{d}|\mathrm{det}(X)|^{-n}d_{\mathrm{a}}X=\int_{\mathrm{Mat}_n{\mathbb{R}}}f(X)\mathrm{d}|\mathrm{det}(X)|^{-n}d_{\mathrm{a}}X=\int_{\mathrm{Mat}_n{\mathbb{R}}}f(XB)\mathrm{d}|\mathrm{det}(X)|^{-n}d_{\mathrm{a}}X$$ for any Haar integrable function $f$ on $\mathrm{Mat}_n(\mathbb{R})$? How can I do it?