$\kappa$-finiteness and $\kappa$-additivity of measures on sets and left invariant measures on discrete groups

Abstract:

For any cardinal $\kappa$ a possibly infinite measure $\mu \geqslant 0$ on a set X is strongly non-$\kappa$-additive if X is partitioned into $\kappa$ or fewer $\mu$-negligible sets. The measure $\mu$ is purely non-$\kappa$-additive if it dominates no nontrivial $\kappa$-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 $\kappa$-finite left invariant measure $\mu$ on a group G of cardinality larger than $\kappa$ is strongly non-$\kappa$-additive. In particular, $\sigma$-finite left invariant measures on infinite groups are strongly finitely additive.