A complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite MV-chain

We consider extensions of the unique additive measure on a finite MV-chain to uncertainty measures on its canonical Girard algebra extension. If the underlying MV-chain has more than two non-trivial elements, in a previous paper we have proved the non-existence of strongly additive measure extensions, where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so-called divisible disjoint unions. This negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all MV-subalgebras of the canonical Girard algebra extension. We obtain a characterization of all such MV-subalgebras which are in fact MV-chains and, using this result, we come up to a characterization of all weakly additive measure extensions. Moreover, we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions, except for underlying MV-chains with one or two or four non-trivial elements.