Following (2) we say that a measure μ on a ring
is semifinite if
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00054778/resource/name/S0008414X00054778_eqn01.gif?pub-status=live)
Clearly every σ-finite measure is semifinite, but the converse fails.
In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring
that possess unique extensions to the σ-ring
generated by
(Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.