Order-theoretic properties and separability of some sets of quasi-measures

Abstract

Abstract Let $${{\mathfrak {M}}}$$ M and $${{\mathfrak {R}}}$$ R be algebras of subsets of a set $$\Omega $$ Ω with $${\mathfrak {M}}\subset {\mathfrak {R}}$$ M ⊂ R . Denote by $$E(\mu )$$ E ( μ ) the set of all quasi-measure extensions of a given quasi-measure $$\mu $$ μ on $${\mathfrak {M}}$$ M to $${\mathfrak {R}}$$ R . We present some results on the coincidence of the bands, in $${\mathfrak {R}}$$ R , generated by $$E(\mu )$$ E ( μ ) and extr $$E(\mu )$$ E ( μ ) . Moreover, we show that if $$\mu $$ μ is atomic, then $$E(\mu )$$ E ( μ ) is contained in a principal band in $${\mathfrak {R}}$$ R if and only if it is separable. Another sufficient condition for the separability of $$E(\mu )$$ E ( μ ) is also presented.