Typical upper [formula omitted]-dimensions of measures for [formula omitted

Abstract

For a probability measure μ on a subset of R d , the lower and upper L q -dimensions of order q ∈ R are defined by D ̲ μ ( q ) = lim inf r ↘ 0 log ∫ μ ( B ( x , r ) ) q − 1 d μ ( x ) − log r , D ¯ μ ( q ) = lim sup r ↘ 0 log ∫ μ ( B ( x , r ) ) q − 1 d μ ( x ) − log r . In previous work we studied the typical behaviour (in the sense of Baire's category) of the L q -dimensions D ̲ μ ( q ) and D ¯ μ ( q ) for q ⩾ 1 . In the present work we study the typical behaviour (in the sense of Baire's category) of the upper L q -dimensions D ¯ μ ( q ) for q ∈ [ 0 , 1 ] .