The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

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 $\mathbb {A}$ .

We investigate Talagrand’s construction of an exhaustive pathological submeasure. We consider the forcing associated with this submeasure and we also begin an effort to explicitly describe this construction.

Let B be a complete ccc Boolean algebra and let $\tau_s$ be the topology on B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonempty open set in $(B,\tau_s)$ is topologically dense.

2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.

3. The topological space $(B,\tau_s)$ is sequentially compact if and only if the generic extension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but not independent reals.

Let L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additive measure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measure v and a weakly purely finitely additive measure ξ. We give sufficient conditions for this Yosida-Hewitt-type decomposition to be unique.

A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.

The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

It is shown that the measure algebra on the space 22ω is σ-n-linked for each n ∊ ω.

The concepts of repleteness and more generally measure repleteness are investigated for set-theoretic lattices on specific subspaces of a lattice space. These general results are then applied to specific topological spaces, and we obtain as special cases some known theorems as well as some new results concerning for example, ∝-completeness, realcompactness, measure compactness and Borel-measure compactness.