For the class of almost additive sequences, we establish it conditional variational principle for the dimension spectra in the context of the nonadditive thermodynamic formalism. This generalizes the classical thermodynamic formalism, by replacing the topological pressure of a single function by the topological pressure of a sequence of functions. In particular. we show that each level set of the multifractal decomposition carries a full measure, that is, ail ergodic invariant measure with dimension equal to the dimension of the level set. We also show that the spectra are continuous and that the irregular sets have full dimension. (c) 2009 Elsevier Masson SAS. All rights reserved.