On the measurability of a numerical function with respect to a family of sets

TL;DR

Greco introduces a notion of measurability of nonnegative functions relative to a family of sets (a paving) and defines ${\mathcal{E}}$-measurability via the invariance of the monotone-set-function integral $\int_X f\,d\alpha$ when $\alpha$ and $\beta$ agree on ${\mathcal{E}}$, establishing a threshold-based characterization and density by ${\mathcal{S}}^+(X,{\mathcal{E}})$. The work develops stability, closure properties, and limit behavior of ${\mathcal{M}}^+(X,{\mathcal{E}})$, linking these to semi-compactness of the paving and ultrafilter descriptions when ${\mathcal{E}}$ is an algebra. It then extends the framework to two pavings ${\mathcal{K}}$ and ${\mathcal{U}}$, proving interpolation results (via Lemma 3 and Theorems 4–5) that relate measurability across pavings to topological normality and Stone-space representations. Overall, the paper unifies several notions of measurability (including classical Carathéodory-type, ultrafilter limits, and two-paving normality) within a single integral-based perspective on pavings. The results have implications for understanding how function-measurability interacts with the geometry of set families and their associated algebras.

Abstract

The following document is a translation (from French to English) of: Gabriele H. Greco, Sur la mesurabilité d'une fonction numérique par rapport à une famille d'ensembles, Rendiconti del Seminario Matematico della Università di Padova}, tome 65 (1981), pp. 163--176. Translated by: Jonathan M. Keith, School of Mathematics, Monash University, jonathan.keith@monash.edu. With thanks to: Prof. Andrea D'Agnolo, Editor-in-Chief of the above journal, for permission to publish this translation.