Filter convergence and decompositions for vector lattice-valued measures

Filter convergence of vector lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the σ-additive case is studied, without particular assumptions on the filter; later the finitely additive case is faced, first assuming uniform s-boundedness (without restrictions on the filter), then relaxing this condition but imposing stronger properties on the filter. In order to obtain the last results, a Schur type convergence theorem is used.