Let Xx, X2 be lattices of subsets of a set X with X, c X2. The main result of this paper states that every semifinite tight set function on Xx can be extended to a semifinite tight set function on JT2. Furthermore, conditions assuring that such an extension is uniquely determined or o-smooth at are given. Since a semifinite tight set function defined on a lattice Jf"[and being o-smooth at ] can be identified with a semifinite ^regular content [measure] on the algebra generated by X, the general

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... results are applied to various extension problems in abstract and topological measure theory. Introduction. The construction of a (finitely or countably additive) measure usually proceeds in such a way that one first defines a set function on a small family of sets and thereafter applies a suitable measure extension procedure. One condition which guarantees a measure extension is the concept of tightness that has been studied by several authors (see [18, 32,33, 17, 2] for the case of [0, oo]-valued set functions to which this paper is restricted). The central subject of this paper is the extension of a given tight set function defined on a lattice Jf", of subsets of a set X to a tight set function on a lattice JT2 d Jfx. The main result (2.5) states that every semifinite tight set function X on Jfx can be extended to a Jf2(\)-regular tight set function p on Jf2, where Jf2(A) denotes the family of all Jf2-sets which are subsets of Jf,-sets of finite A-measure. In addition, we give conditions assuring that such an extension p is uniquely determined (2.7) or a-smooth at (2.8). These results are applied to various extension problems in (topological) measure theory. Among others, we obtain a generalization of the main extension theorem of Bachman and Sultan [3-5] (see 3.2). Some further applications deal with the existence of a preimage measure, the extension of the product of two a-finite Baire [Borel] measures to a Baire [Borel] measure on the product space, and the extension of Baire to Borel measures (see 3.8,3.10,3.13,3.14). In particular, we obtain a variant of a measure extension theorem due to Mafik [25]. This result enables us to introduce a class of topological spaces X (including, in particular, all normal metacompact ones) for which the following two statements are valid: X is measure-compact iff each closed discrete subset of X has cardinal of measure zero. X is realcompact iff each closed discrete subset of X has nonmeasurable cardinal. This result (see 3.16) leads to a common generalization of results in [16, 26, 13, 7, 8] .