_k-finiteness and k-additivity of measures on sets and left invariant measures on discrete groups

For any cardinal k a possibly infinite measure μ > 0 on a set X is strongly nonadditive if X is partitioned into k or fewer ii-negligible sets. The measure μ is purely non-K-additive if it dominates no nontrivial K-additive measure. The properties and relationships of these types of measures are examined in relationship to measurable ideal cardinals and real-valued measurable cardinals. Any K-finite left invariant measure μ on a group G of cardinality larger than k is strongly non-K-additive. In particular, σ-finite left invariant measures on infinite groups are strongly finitely additive.