On the average L^q-dimensions of typical measures belonging to the Gromov–Hausdorff–Prohoroff space. The limiting cases: q = 1 and q = ∞

Abstract

Abstract. We study the averageL-dimensions of typical Borel probability measures belonging to the Gromov–Hausdorff–Prohoroff space (of all Borel probability measures with compact supports) equipped with the Gromov–Hausdorff–Prohoroff metric. Previously the lower and upper average L-dimensions of a typical measure μ have been found for q ∈ (1,∞). In this paper we determine the lower and upper average L-dimensions of a typical measure μ in the two limiting cases: q = 1 and q = ∞. In particular, we prove that a typical measure μ is as irregular as possible: for q = 1 and q = ∞, the lower average L-dimension attains the smallest possible value, namely 0, and the upper average L-dimension attains the largest possible value, namely ∞. The proofs rely on some non-trivial semi-continuity properties of L-dimensions that may be of interest in their own right.