Typical upper L-q-dimensions of measures for q is an element of [0,1]

For a probability measure mu on a subset of R-d, the lower and upper L-q-dimensions of order q epsilon R are defined by

(D) under bar (mu)(q) = lim(r SE arrow 0)inf log integral mu(B(x, r))(q-1)d mu(x)/-logr,

(D) over bar (mu)(q) = lim(r SE arrow 0)sup log integral mu(B(x, r))(q-1)d mu(x)/-logr.

In previous work we studied the typical behaviour (in the sense of Baire's category) of the L-q-dimensions (D) under bar (mu)(q) and (D) over bar (mu)(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 SAS. (C) 2007 Elsevier Masson SAS. All rights reserved.