On Marczewski-Burstin Like Characterizations of Certain σ-Algebras and σ-Ideals

Abstract

Consider a $\sigma$-ideal, $\sigma$-algebra pair $\cal{I} \subseteq \cal{A}$ on a Polish space $X$ which has no isolated points, such that $\cal{A}$ contains all the Borel subsets of $X$ while $\cal{I}$ contains all the countable subsets of $X$, but none of the perfect subsets of $X$. We show that if $(\cal{I}, \cal{A})$ admits a simultaneous MB-like characterization consisting of Borel sets, then $(\cal{I},\cal{A})$ is $((s_{0}), (s))$, the $\sigma$-ideal, $\sigma$-algebra pair of Marczewski null, Marczewski measurable sets. We deduce some results about uniformly completely Ramsey sets.