Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property

Abstract

Improving a result of M. Talagrand, under the assumption of a weak form of Martin's axiom, we construct a totally disconnected compact Hausdorff space $K$ such that the Banach space $C(K)$ of continuous real-valued functions on $K$ is a Grothendieck space but there exists a sequence $(\mu_n)$ of Radon measures on $K$ such that $\mu_n(A)\to0$ for every clopen set $A\subseteq K$ and $\int_Kfd\mu_n\not\to0$ for some $f\in C(K)$. Consequently, we get that Martin's axiom implies the existence of a Boolean algebra with the Grothendieck property but without the Nikodym property.