Selection Principles for Measurable Functions and Covering Properties

Abstract

Let $\mathcal{A}\subset \mathcal{P}(X)$, $\varnothing ,X\in \mathcal{A}$, $\mathcal{A}$ 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 $F\subseteq {}^{X}R$ of real functions. The main result of the paper Theorem reads as follows: if~$\Phi =\Omega ,\Gamma$, then for any couple $⟨\Phi ,\Psi ⟩$ different from $⟨\Omega ,\mathcal{O}⟩$, $X$ 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}${}_{fin}$ and {\rm U}${}_{fin}$. Some related results are also presented.

21 pages