  Filter convergence of vector lattice-valued measures is considered, in order
to deduce theorems of convergence for their decompositions. First the
$\sigma$-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.
