Revisiting the multifractal analysis of measures

New proofs of theorems on the multifractal formalism are given. They yield results even at points q for which Olsen's functions b(q) and B(q) differ. Indeed, we provide an example of a measure for which the functions b and B differ and for which the Hausdorff dimensions of the sets Xα (the level sets of the local Hölder exponent) are given by the Legendre transform of b and their packing dimensions by the Legendre transform of B. © European Mathematical Society.