If X is a separable non-reflexive Banach space, then the set NA of all norm-attaining elements of X* is not a w*-Gδ subset of X*. However if the norm of X is locally uniformly rotund, then the set of norm attaining elements of norm one is w*-Gδ. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If X is separable and non-reflexive, there exists an equivalent Gâteaux-smooth norm on X such that the set of all Gâteaux-derivatives is not norm-Borel.

A natural rank function is defined on the set DS of everywhere divergent sequences of continuous functions on the unit circle T. The rank function provides a natural measure of the complexity of the sequences in DS, and is obtained by associating a well-founded tree with each such sequence. The set DF of everywhere divergent Fourier series, and the set DT of everywhere divergent trigonometric series with coefficients that tend to zero, can be viewed as natural subsets of DS. It is shown that the rank function is a coanalytic norm which is unbounded in ω1 on DF. From this it follows that DF, DT and DS are not Borel subsets of the Polish space SC(T) of all sequences of continuous functions on T. Finally an alternative definition of the rank function is formulated by using nested sequences of closed sets.

The main results of this paper are the following: (1) An extension theorem for a uniform semigroup-valued measure on a ring to the generated σ-ring. This result unifies the classieal Hahn-Carathéodory theorem, the extension theorem of Sion and a more recent result of Weber.

(2) A theorem stating that every monocompact additive uniform semigroup-valued set function on a semiring is σ-additive. This result generalizes several earlier theorems of Alexandroff, Dinculeanu-Kluvanek, Glicksberg, Huneycutt, Mallory, Marczewski, Millington and Topsøe.