It is stated as a fact in this correction paper that
if $A$ is a subspace [and $U(A)$ is the set of unit vectors in $A$] then there is a unique probability measure defined on $U (A)$ which is invariant under any unitary transformation of $A$, which we call the uniform distribution on $U(A)$
Could anyone point me to a resource elaborating on this? This is mainly a graph theory paper which I am trying to digest, but I haven't taken a course in measure theory. I would like to know how to construct this measure/what it looks like.
Here is one approach to constructing such a measure. $A$ is isomorphic to $\mathbb{R}^k$ for some $k$, so we can identify $U(A)$ with the unit sphere $S^{k - 1} \subset \mathbb{R}^k$. $S^{k - 1}$ is a $C^1$ $k - 1$ dimensional surface, so there is the natural surface measure $\nu$ on $S^{k - 1}$ such that if $O \subset \mathbb{R}^{k - 1}$ is open and $\phi \colon O \to U \subset S^{k - 1}$ is a coordinate chart, then for any Borel measurable $f \colon S^{k - 1} \to [0, \infty]$, $$\int_{U}f(y)\,d\nu(y) = \int_{O}f(\phi(x))\sqrt{g(x)}\,dx,$$ where $g(x) = \det(D\phi(x)^*D\phi(x))$. It is not difficult to deduce from the above formula that $\nu$ is invariant under unitary transformations. The desired probability measure would be $\nu' = \frac{1}{\nu(S^{k - 1})}\nu$.