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
Famous quotes containing the word measure:
“I thought of rhyme alone,
For rhyme can beat a measure out of trouble
And make the daylight sweet once more....”
—William Butler Yeats (1865–1939)
“What is life but the angle of vision? A man is measured by the angle at which he looks at objects. What is life but what a man is thinking all day? This is his fate and his employer. Knowing is the measure of the man. By how much we know, so much we are.”
—Ralph Waldo Emerson (1803–1882)