Associated Measure
When V is n-dimensional Euclidean space, one may define a uniform measure on Gr(r, n) in the following way. Let θn 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, γr, n (g A) = γr, n (A) for all g in O(n). Since θn (O(n))=1, we have γr, n (Gr(r, n))=1. Moreover, γr, 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.
Read more about this topic: Grassmannian
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