Selection Principles for Measurable Functions and Covering Properties

Let ${\mathcal A}\subset {\mathcal P}(X)$, $\emptyset, X\in {\mathcal A}$, ${\mathcal A}$ being closed under finite intersections. If $ψ={o},ω,γ$, then $Ψ({\mathcal A})$ is the family of those $ψ$-covers ${\mathcal U}$ for which ${\mathcal U}\subseteq {\mathcal A}$. In~\cite{BL2} I have introduced properties $(Ψ_0$ of a~family $F\subseteq {}^XR$ of real functions. The main result of the paper Theorem reads as follows: if~$Φ=Ω,Γ$, then for any couple $\langle Φ,Ψ\rangle$ different from $\langle Ω,{\mathcal O}\rangle$, $X$ has the covering property~{\rm S}${}_1(Φ({\mathcal A}),Ψ({\mathcal A}))$ if and only if the family of non-negative upper ${\mathcal A}$-semimeasurable real functions satisfies the selection principle~{\rm S}${}_1(Φ_0,Ψ_0)$. Similarly for {\rm S}${}_{\scriptstyle fin}$ and {\rm U}${}_{\scriptstyle fin}$. Some related results are also presented.

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