Constructing an Elementary Measure on a Space of Projections

Bypassing much theory from integral geometry, we construct an elementary measure on a space whose elements can represent rank k orthogonal projections in N. By replacing the Grassmannian GN k with a simple product space k j=1 SN−1 we are able to reproduce certain important features of the nontrivial measure on GN k invariant under the action of the orthogonal group (a property also enjoyed by our construction). As a motivating example we show that our construction enables the proof of a recent embedding theorem due to Foias and Olson to be completed using only standard methods of analysis