Selection Principles for Measurable Functions and Covering Properties

TL;DR

This paper explores selection principles for measurable functions and covering properties, establishing equivalences between covering properties of sets and selection principles for families of functions, with implications for measure theory.

Contribution

It introduces new equivalences between covering properties of sets and selection principles for families of functions, extending previous work in measure theory and topology.

Findings

Characterizes when certain covering properties hold for sets.

Establishes equivalences between set covering properties and function selection principles.

Provides related results in the context of measurable functions.

Abstract

Let ${\mathcal A}\subset {\mathcal P}(X)$, $\emptyset, X\in {\mathcal A}$, ${\mathcal A}$ being closed under finite intersections. If $\psi={o},\omega,\gamma$, then $\Psi({\mathcal A})$ is the family of those $\psi$-covers ${\mathcal U}$ for which ${\mathcal U}\subseteq {\mathcal A}$. In~\cite{BL2} I have introduced properties $(\Psi_0$ of a~family $F\subseteq {}^XR$ of real functions. The main result of the paper Theorem reads as follows: if~$\Phi=\Omega,\Gamma$, then for any couple $\langle \Phi,\Psi\rangle$ different from $\langle \Omega,{\mathcal O}\rangle$, $X$ has the covering property~{\rm S}${}_1(\Phi({\mathcal A}),\Psi({\mathcal A}))$ if and only if the family of non-negative upper ${\mathcal A}$-semimeasurable real functions satisfies the selection principle~{\rm S}${}_1(\Phi_0,\Psi_0)$. Similarly for {\rm S}${}_{\scriptstyle fin}$ and {\rm U}${}_{\scriptstyle fin}$. Some related results are also presented.