Let $$\operatorname{Aff}_n(\mathbb{R}) := \left\{\begin{pmatrix} A & v\\ 0 & 1 \end{pmatrix}: \, A \in \operatorname{GL}_n(\mathbb{R}), v \in \mathbb{R}^n\right\}$$ be the group of affine transformations on $\mathbb{R}^n$.
How can I find an explicit formula for a Haar measure on this group?
Up to now, I've been given Haar measures and was (mostly) able to verify they were indeed Haar measures, but I have no idea where I would start to construct one. We also skipped the proof of existence of a Haar measure since apparently it is not very enlightening.
Denote $dA$ the standard Lebesgue measure on $\mathcal{M}_n(\mathbb{R})$, that is
$$dA = \prod_{1\le i,i \le n} da_{i,j}$$
where $A = (a_{i,j})_{1\le i,i \le n}$ , and similarly $d\nu = \prod_{i = 1}^n \nu_i$
The formula for the change of variables in an integral can be formally denoted as $d(A'A) = \left|det A'\right|^n \, dA$ if $A'$ is a fixed matrix. And similarly, of $A'$ is a fixed matrix and $\nu'$ a fixed vector, we have $d(A'\nu + \nu') = \left|det A'\right| \, d\nu$. Then I think that taking
$$d^*\left(\begin{matrix}A & \nu\\0&1\end{matrix}\right) = \frac{dA \cdot d\nu}{\left|\det A\right|^{n+1}}$$
gives you a left invariant measure on $\operatorname{Aff}_n(\mathbb{R})$, as you can check that
$$\begin{align*} d^*\left[ \left(\begin{matrix}A' & \nu'\\0&1\end{matrix}\right)\left(\begin{matrix}A & \nu\\0&1\end{matrix}\right)\right] &= d^*\left[ \left(\begin{matrix}A'A & A'\nu+ \nu'\\0&1\end{matrix}\right)\right]\\ &= \frac{d(A'A) \cdot d(A'\nu+\nu')}{\left|det(A'A)\right|^{n+1}} \\ &= \frac{\left|\det A'\right|^n\, dA\cdot \left|det A'\right|\, d\nu}{\left|\det A'\right|^{n+1} \, \left|\det A\right|^{n+1}} \\ &= \frac{dA\cdot d\nu}{\left|\det A\right|^{n+1}} \\ &= d^*\left(\begin{matrix}A & \nu\\0&1\end{matrix}\right) \end{align*}$$
In general, when elements are parametrized by some $X \in \mathbb{R}^N$, the Haar measure will often be of the form $f(X) \, dX$ for some function $f$, where $dX$ denotes the usual Lebesgue measure. Finding the measure, boils down to finding this function $f$.