Typical L q -dimensions of measures

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

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We study the typical behaviour (in the sense of Baire's category) of the L-q-dimensions D-mu(q) and D-mu(q). We prove that a typical measure mu is as irregular as possible: for all q >= 1, the lower L-q-dimension D-mu(d) attains the smallest possible value and the upper L-q-dimension D-mu(d) attains the largest possible value.