Typical upper Lq-dimensions of measures for q∈[0,1]

AbstractFor a probability measure μ on a subset of Rd, the lower and upper Lq-dimensions of order q∈R are defined byD̲μ(q)=lim infr↘0log∫μ(B(x,r))q−1dμ(x)−logr,D¯μ(q)=lim supr↘0log∫μ(B(x,r))q−1dμ(x)−logr. In previous work we studied the typical behaviour (in the sense of Baire's category) of the Lq-dimensions D̲μ(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 Lq-dimensions D¯μ(q) for q∈[0,1]