The classes of tight, r-additive, and a-additive linear functional on the double centraliser algebra of a C*-algebra A are defined. The algebra A is called measure compact if all three classes coincide. Several theorems relating the existence of certain types of approximate identities in A to measure compactness of A are proved. Next, permanence properties of measure compactness are studied. For example, the C*-algebra tensor product of two measure compact C*-aIgebras is measure compact. Next,

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... he question of weak-star metrizability of the positive cone in the space of tight measures is considered. In the last part of the paper, another topology is defined and is used to study the relationship of measure compactness of A and the property that the strict topology is the Mackey topology in the pairing of M(A) with the tight functionals on M(A). Also, in the last section of the paper is an extension of a result of Glickberg about finitely additive measures on pseudocompact topological spaces.