In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.

Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Hölder exponents 0 and ∞.