This paper investigates the eventwise aggregations of decomposable mea
sures preserving the same decomposable property. These operations are
obtained by solving a functional equation closely related to the bisym
metry property. Known results for probability as well as possibility m
easures can be derived as particular cases of our approach. In additio
n, the unicity of weighted consensus functions is proved in the Archim
edean case. An extension of Von Neumann-Morgenstern utility theory is
outlined, where probabilities are changed into decomposable measures.