Two modifications of the Suslin hypothesis serve as a motivation of the paper:
(H1) every ccc compact space which cannot be mapped continuously onto \([0,1]^{\omega_1}\) is separable; and
(H2) every compact space supporting a measure which cannot be mapped continuously onto \([0,1]^{\omega_1}\) is separable.
It is a result by S. Todorcevic [Topology Appl. 101, No. 1, 45–82 (2000; Zbl 0973.54004)] that (H1) does not hold in ZFC. On the other hand, it is an open question whether the weaker condition (H2) is consistent with ZFC. In this context, a natural question is whether the space from Todorčević’s construction can support a measure. The authors examine spaces constructed by Todorčević’s technique from the measure-theoretic view. Under the assumption \(\mathrm{add}(\mathcal N)=\mathrm{non}(\mathcal M)\), they modify Todorčević’s construction and get a counterexample for (H2).
Let \(K\) be a compactification of \(\omega\). The subspace \(\{f\in C(K):f(x)=0\) for \(x\in K\setminus\omega\}\) of \(C(K)\) is called the natural copy of \(c_0\) in \(C(K)\). The authors prove that there is a compactification \(L\) of \(\omega\) such that \(L\setminus\omega\) is non-separable and supports a measure and the natural copy of \(c_0\) in \(C(L)\) is complemented in \(C(L)\), i.e., it is a projection of \(C(L)\). If \(\mathrm{add}(\mathcal N)=\mathrm{non}(\mathcal M)\), then additionally it can be required that \(L\setminus\omega\) does not map continuously onto \([0,1]^{\omega_1}\).
The authors also discuss examples of spaces not supporting measures but satisfying quite strong chain conditions. The main tool in the paper relies on the A. Kamburelis’s characterization of Boolean algebras [Arch. Math. Logic 29, No. 1, 21–28 (1989; Zbl 0687.03032)] supporting a measure as those Boolean algebras that are \(\sigma\)-centered in some random real extensions.