ON THE AVERAGE Lq-DIMENSIONS OFTYPICAL MEASURES BELONGING TO THEGROMOV HAUSDORFF PROHOROFF SPACE.THE LIMITING CASES q = 1 AND q = ∞

摘要

We study theaverage Lq-dimensionsof typical Borel probability measuresbelonging to the Gromov Hausdorff Prohoroff space (of all Borel probability measureswith compact supports)equipped with the Gromov Hausdorff Prohoroff metric.Previously thelower and upperaverageLq-dimensionsof a typical measure have been found for q (1, ).In this paperwe determinethe lower and upper average Lq-dimensionsof a typical measure in the two limiting cases: q = 1 and q = .In particular, we prove that a typical measure is as irregular aspossible:for q = 1 and q = ,the lower average Lq-dimensionattains the smallest possible value, namely 0,andthe upper average Lq-dimensionattains the largest possible value, namely .The proofs rely on some non-trivial semi-continuity propertiesof Lq-dimensionsthat may be of interest in their own right.