{'fraction_non_alphanumeric': 0.016566265060240965, 'fraction_numerical': 0.0030120481927710845, 'mean_word_length': 4.833333333333333, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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 = 1kSN − 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.