Published online by Cambridge University Press: 13 September 2021
For given Boolean algebras $\mathbb {A}$ and
$\mathbb {B}$ we endow the space
$\mathcal {H}(\mathbb {A},\mathbb {B})$ of all Boolean homomorphisms from
$\mathbb {A}$ to
$\mathbb {B}$ with various topologies and study convergence properties of sequences in
$\mathcal {H}(\mathbb {A},\mathbb {B})$. We are in particular interested in the situation when
$\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on
$\mathbb {A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on
$\mathcal {H}(\mathbb {A},\mathbb {B})$ for a Boolean algebra
$\mathbb {B}$ carrying a strictly positive measure and convergence properties of sequences of measures on
$\mathbb {A}$.