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