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 ${\mu }_n(A)\to 0$ for every clopen set $A\subseteq K$ and ${\int }_{K}fd{\mu }_n\nrightarrow 0$ 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.

arXiv commentNew references added, minor corrections