Denote by $G(n,k)$ the real Grassmannian, the set of $k$-dimensional subspaces of $\mathbb{R}^n$. It is a topological space, even metrizable (see A metric for Grassmannians), and so it is a measurable space with the Borel $\sigma$-algebra.

Here are two ways to furnish a probability measure on $G(n,k)$:

1) Pass the Haar probability measure on $O(n)$ to $G(n,k)$ via its natural action on $G(n,k)$ (see Measure on real Grassmannians),

2) Let $R = \mathbb{R}^{n-k} \times \{0\}^k$ be the subspace consisting of vectors ending in $k$ zeros (in the standard basis), and let $G^*(n,k) = \{V \in G(n,k) \mid V \cap R = \{0\} \}$. Consider the (standard?) coordinate chart whereby a real $(n-k) \times k$ matrix $M$ is associated with the element of $G^*(n,k)$ spanned by the columns of the matrix $\left(\begin{array}{c} M \\ \hline \text{Id}_{k \times k} \end{array}\right)$. Finally, pass your favorite probability measure on $\mathbb{R}^{(n-k)k}$ which is equivalent to the Lebesgue measure (say, the Gaussian measure) to $G^*(n,k)$ via this chart, and extend it to $G(n,k)$ by declaring that $G(n,k) \setminus G^*(n,k)$ has zero measure.

I believe these two measures on $G(n,k)$ are equivalent (in the usual sense that they share the same null sets), but I don't know how to show it. Any ideas would be much appreciated.