TITLE: Haar measure on the groups SO(n) and SO(n,m)
QUESTION [8 upvotes]: Would you please give me some information about Haar measure on special orthogonal group SO(n) and indefinite special orthogonal group SO(n,m)?
Thank you so much!

REPLY [5 votes]: The Haar measure on $SO(n)$ can be described inductively as follows.
Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis for $\mathbb{R}^n$, and let $H$ be the stabsubgroup of $SO(n)$ that fixes $e_n$.  Note that $H$ is isomorphic to $SO(n-1)$, so it has a Haar measure, which we already know.
Now, if $g_1,g_2\in SO(n)$, then $g_1H = g_2H$ if and only if $g_1(e_n)=g_2(e_n)$.  Thus, the coset space $SO(n)/H$ is isomorphic to the sphere $S^{n-1}$.  This sphere also has a nice, well-known measure.
Now, let $f\colon S^{n-1}\to SO(n)$ be any measurable function that picks a transversal for each coset.  That is, for any point $p$ in the sphere, $f(p)$ should be an element of $SO(n)$ that maps $e_n$ to $p$.  This transversal need not be continuous---only measurable.  Then the measure of a measurable subset $X\subseteq SO(n)$ is given by the formula
$$
\mu_{SO(n)}(X) \;=\; \int_{S^{n-1}} \mu_{H}\bigl(f(p)^{-1}(X\cap f(p)H)\bigr)\; d\mu_{S^{n-1}}(p)
$$
where $\mu_H$ denotes the Haar measure on $H$, and $\mu_{S^{n-1}}$ denotes the standard measure on $S^{n-1}$.
Is that helpful?  There is (presumably) an alternative description involving the Jacobian of the exponential map $\mathfrak{so}(n) \to SO(n)$, if that would be better.
I don't know much about the Haar measure for $SO(n,m)$ off the top of my head, but there should be an inductive description for it similar to the description above.