Measurable functions on σ-frames

Abstract

We study (semi)measurable functions on σ-frames, extending the theory of real-valued functions from frames to σ-frames. The new objects of study are the σ-frame homomorphisms ▪ from the usual frame of reals into the congruence lattice of a σ-frame L, and its subclass of measurable functions L(R)→L. The desired extension faces two obstacles: (1) in general, σ-frames have no uncountable joins and are not pseudocomplemented; (2) σ-sublocales, that is, the subobjects in the dual category of the category of σ-frames and σ-frame homomorphisms, cannot be described as concrete subsets of the σ-frame L, unlike their counterpart in the category of locales, forcing us to work in the congruence lattice of L.Nevertheless, it is shown that, despite (1), the familiar method for generating real functions on frames via scales can be extended to arbitrary σ-frames. This is achieved by the new notions of σ-scale and finite σ-scale. It is also shown that, despite (2), the familiar insertion, extension and separation results for real-valued functions in several classes of frames (normal, extremally disconnected, G-perfect, F-perfect, perfectly normal) can be proved without involving uncountable joins and pseudocomplements, thus allowing their extension to measurable and semimeasurable functions.