Grassmannian - Associated Measure

Associated Measure

When V is n-dimensional Euclidean space, one may define a uniform measure on Gr(r, n) in the following way. Let θ_n be the unit Haar measure on the orthogonal group O(n) and fix V in Gr(r, n). Then for a set A ⊆ Gr(r, n), define

This measure is invariant under actions from the group O(n), that is, γ_{r, n} (g A) = γ_{r, n} (A) for all g in O(n). Since θ_n (O(n))=1, we have γ_{r, n} (Gr(r, n))=1. Moreover, γ_{r, n} is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.