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For any cardinal K a possibly infinite measure ,u > 0 on a set X is strongly non-K-additive if X is partitioned into K or fewer ,u-neghigible sets. The measure ,u 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 ,u on a group G of cardinality larger than K iS strongly non-K-additive. In particular, a-finite left invariant measures on infinite groups are strongly finitely additive. In [2], P. Erdos and R. D. Mauldin showed that the only countably additive, a-finite, left invariant, positive measure on all the subsets of a group G is the trivial measure. One immediate generalization which suggests itself is that, for an infinite cardinal K and a group G with card(G) > K, the only K-additive K-finite positive left invariant measure is the trivial measure. We establish a stronger statement, even when K = w, that any K-finite positive measure on such a group G is strongly non-K-additive. DEFINITION 1. Let X be a set and let It > 0 be a finitely additive measure on all subsets of X and let K be a cardinal number. (i) X, or It, is K-finite iff it is the union of K or fewer sets A with 1i(A) < 00. (ii) X is K-negligible and jt is strongly non-K-additive iff X is the union of K or fewer sets A with ,u(A) = 0. (iii) jt is K-additive iff whenever {Aa: a c I) is an increasing family of subsets with card(r) < K then (4U {Aa: a E I)) = sup{ t(Aa): a E '). (iv) ,u is purely non-K-additive iff the only K-additive measure it dominates is 0. DEFINITION 2. A purely non-w-additive measure is called a purely finitely additive measure. A strongly non-w-additive measure is called a strongly finitely additive measure. An c-finite measure is a a-finite measure. Only the trivial measure is both K-additive and purely non-K-additive. Any strongly non-K-additive measure is purely non-K-additive. If Al and ju2 are finitely additive positive measures i-, V A2 defined by P1 V 1i2(A) = sup{t 1 (A') + J2(A \A'): A' c A for A c X, Received by the editors December 12, 1978 and, in revised form, June 13, 1979. 1980 Mathematics Subject Classification. Primary 28A12; Secondary 03E55, 28D99.

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