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 RN . By replacing the Grassmannian GN,k with a simple product space ⊗j=1S 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 & Olson to be completed using only standard methods of analysis.