Correction to: Order-theoretic properties and separability of some sets of quasi-measures

1 Correction to: Ricerche mat. (2018) 67:581–595 https://doi.org/10.1007/s11587-018-0373-0

Here is a corrected version of the Abstract.

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

The corrections consist in replacing “${\mathfrak {R}}$” by “$ba({\mathfrak {R}})$” in lines 3 and 5 thereof.

Here are other corrections needed in the text.

• 1. In the fifth paragraph on p. 585, the final formula should read as follows: $\mu =\sum _{\nu \in {\mathcal U}_\mu }\nu$.

• 2. Condition (i) of Proposition 4 on p. 585 should read as follows: ${ba({\mathfrak {M}})=B_{ult({\mathfrak {M}})}}$.

• 3. In Remark 2 on p. 586, the displayed formula should read as follows:

  $$\begin{aligned} B_{pa({\mathfrak {M}})}=B_{ult({\mathfrak {M}})}. \end{aligned}$$

• 4. The first inequality in the proof of Proposition 5 on p. 586 should read as follows: $\tau \geqslant \sum _{\pi \in {\mathcal F}}\tau _\pi$.

• 5. In Remark 3 on p. 586, “$2^c$” should read as follows: “$2^{\mathfrak {c}}$” twice.

• 6. In the line following the proof of Theorem 3 on p. 592, “equivalence of” should read “equivalent”.

• 7. In the first paragraph on p. 593, “a cardinality${}\ge 1$” should read “a cardinal${{}\ge 1}$”.