Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 647–672
University of St. Andrews,
Department of Mathematics
St. Andrews, Fife KY16 9SS, Scotland; lo 'at' st-and.ac.uk
Abstract. We study the average Lq-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 Lq-dimensions of a typical measure μ have been found for q ∈ (1,∞). In this paper we determine the lower and upper average Lq-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 Lq-dimension attains the smallest possible value, namely 0, and the upper average Lq-dimension attains the largest possible value, namely ∞. The proofs rely on some non-trivial semi-continuity properties of Lq-dimensions that may be of interest in their own right.
2010 Mathematics Subject Classification: Primary 28A78, 28A80.
Key words: Lq-dimension, Gromov–Hausdorff–Prohoroff space, Gromov–Hausdorff–Prohoroff metric, Hölder mean, Cesàro mean, Baire category.
Reference to this article: L. Olsen: On the average Lq-dimensions of typical measures belonging to the Gromov–Hausdorff–Prohoroff space. The limiting cases: q = 1 and q = ∞. Ann. Acad. Sci. Fenn. Math. 45 (2020), 647–672.
https://doi.org/10.5186/aasfm.2020.4535
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