Nonseparable growth of the integers supporting a measure (Q2352121)

It is a consequence of Parovičenko's theorem that under the assumption of the Continuum Hypothesis there is a compactification of the space of integers that has the Stone space of the measure algebra as its remainder. This remainder is non-separable but it does have countable cellularity. In [Topol. Proc. 5, 11--25 (1981; Zbl 0464.54017)] \textit{M. G. Bell} constructed a ZFC example with these properties. It is still an open question whether one can have, in ZFC, a remainder that is non-separable and carries a non-trivial measure; the authors indicate that Bell's example, and others, do not carry measures. They construct a compactification of the space of integers that has a non-separable remainder that does carry a non-trivial measure, though under the assumption that \(\mathfrak{b}=\mathfrak{c}\).