Epic math battle of history: Grothendieck vs Nikodym

Abstract

We define a $\sigma$-centered notion of forcing that forces the existence of a Boolean algebra with the Grothendieck property and without the Nikodym property. In particular the existence of such an algebra is consistent with the negation of the continuum hypothesis. The algebra we construct consists of Borel subsets of the Cantor set and has cardinality $\omega_1$. We also show how to apply our method to streamline Talagrand's construction of such an algebra under the continuum hypothesis.