Selection Principles for Measurable Functions and Covering Properties

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.