  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 $\mathbb{A}$.
