Measure on real Grassmannians OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix a space V in your Grassmannian. Then for any subset A, the measure of A is $$\gamma(A)=\theta(g\in O(n) \mid gV \in A).$$
Fair enough. Two things Wikipedia does not really tell me, though:

*

*Where does this construction originate? I would imagine something like this to be fairly folklore, but it would sure be nice if someone has a reference.


*Does anyone know of especially interesting applications of this measure? Since the Wikipedia article cruelly lacks context, I would really like to see this idea in action.
Thanks in advance!
 A: It's easy to describe the metric that gives rise to this measure: define a map from the Grassmannian of $k$-planes in $\mathbb{R}^n$ to the set of $n$ by $n$ matrices by associating to a $k$-plane $V$ the orthogonal projection $\pi_V$ onto $V$.  This embeds the Grassmannian as a real algebraic subvariety of the space of $n$ by $n$ matrices (characterized as the set of symmetric matrices $\pi$ with trace $k$ such that $\pi^2=\pi$) and there is a natural choice of metric given by $d(V,W)=|\pi_V-\pi_W|$ where $|\cdot|$ denotes the sup norm on the space of $n$ by $n$ matrices.  It follows from the definition that the metric is $O(n)$-invariant and therefore gives rise to an $O(n)$-invariant measure (up to scalars the one you ask about).
A: First of all, a general fact. Any transitive homogeneous space $X$ of a
compact group $K$ has a unique $K$-invariant probability measure. Existence:
take the image $\nu$ of the normalized Haar measure $m$ on $K$ under the map
$g\mapsto gx_0$ (that's the construction you refer to). The measure $\nu$ is
well-defined, since the mass of $m$ is finite, it does not depend on $x_0\in
X$ by right invariance of $m$, and is $K$-invariant by left invariance of $m$.
Uniqueness: take an arbitrary $K$-invariant measure $\nu'$ on $X$, and
consider its convolution $m\ast\nu'$ with the measure $m$ (i.e., the image of
the product of $m$ and $\nu'$ under the map $(g,x)\mapsto gx$). Then, on one
hand $m\ast\nu'=\nu'$ by $K$-invariance of $\nu'$, on the other hand
$m\ast\nu'=\nu$ by the above construction.
Thus, since the Grassmannian in question has a transitive compact group of
automorphisms, it carries a "natural" invariant measure. So that
"platonically" it is always there - like, for instance, the Riemannian volume
on a Riemannian manifold (by the way, as mentioned before, any invariant
Riemannian metric on the Grassmannian produces the measure in question in this
way).
However, there is one subtlety here which has so far remained unnoticed. In
order to define the Grassmannian one needs a linear structure, whereas the
orthogonal group $O(n)$ is defined in terms of the Euclidean structure.
Therefore, the "canonical" measure we are talking about is only canonical with
respect to the given Euclidean structure on the linear space $V=R^n$. So, if
we look at the problem from this point of view, we obtain a map which assigns
to any Euclidean structure on $V$ a probability measure on the Grassmannian
$Gr_k(V)$. In fact, this measure depends only on the projective class of the
Euclidean structure (i.e., on the corresponding similarity structure), which
are parameterized by the Riemannian symmetric space $S=SL(n,R)/SO(n)$
(equivalently, one can say that we consider only the Euclidean structures on
$V$ with the same volume form, whence $SL$ instead of $GL$).
Thus, we have a map $x\mapsto\nu_x$ from $S$ to the space $P(Gr_k)$ of
probability measures on $Gr_k$. One can show that this map is an injection, so
that it can be used in order to compactify the symmetric space $S$ by taking
its closure in the weak$^*$ topology of $P(Gr_k)$. This is an example of a
so-called Satake-Furstenberg compactification, which can be defined for an
arbitrary non-compact Riemannian symmetric space. In the case of the space
$S=SL(n,R)/SO(n)$ all such compactifications are obtained by considering
rotation invariant measures on the flag space of $V$ and its equivariant
quotients (in particular, Grassmannians). In the general case the role of the
flag space is played by the so-called Furstenberg boundary, which is the
quotient of the semi-simple Lie group by its minimal parabolic subgroup. The
most recent reference for all this is the book by Borel and Ji.
The simplest non-compact symmetric space is the hyperbolic plane. In this case
the Furstenberg boundary (the associated "flag space") is just the boundary
circle in the disk model. Each point of the hyperbolic plane determines a
unique probability measure on the boundary circle invariant with respect to
the rotations around this point. These measures appear in the classical
Poisson formula for bounded harmonic functions in the unit disk (usually it is
written in terms of just a single measure corresponding to the Euclidean
center of the disk; the other measures appear in the guise of their
Radon-Nikodym derivatives with respect to this one, which is precisely the
Poisson kernel).
A: This measure is the unique $O(n)$ invariant measure on the Grassmannian up to a multiplication by a scalar. The following lecture note explains invariant measures on homogeneous spaces. One application is in harmonic analysis on homogeneous spaces, see for example: the following review article
A: A nice application is a Crofton-like formula for the codimension $k=\dim V$ submanifolds of $S^{n-1}\subset\mathbb{R}^n$. If $X$ is such a submanifold, then its $n-k-1$ dimensional measure is the average number of points of $V\cap X$, $V\in\mathrm{Gr}_{n,k}$, with respect to the said measure, multiplied by half the measure of $S^{n-k-1}$ (which has 2 intersection point for almost all $V$).
There also is an affine version with the grassmannian of $k$-dimensional affine subspaces of (euclidean) $\mathbb{R}^n$ and codimension $k$ submanifolds of $\mathbb{R}^n$
(the original Crofton formula is the case $n=2$, $k=1$).
Appropriate keywords would be integral geometry, and perhaps also geometric measure theory.
