On sequences of homomorphisms into measure algebras and the Efimov Problem

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 $\mathbb{A}$.