The induced capacity and Choquet integral monotone convergece

Given a probability measure over a state space, a partial collection (sub-$\sigma$-algebra) of events whose probabilities are known, induces a capacity over the collection of all possible events. The \emph{induced capacity} of an event $F$ is the probability of the maximal (with respect to inclusion) event contained in $F$ whose probability is known. The Choquet integral with respect to the induced capacity coincides with the integral with respect to a \emph{probability specified on a sub-algebra} (Lehrer \cite{Lehrer2}). We study Choquet integral monotone convergence and apply the results to the integral with respect to the induced capacity. The paper characterizes the properties of sub-$\sigma$-algebras and of induced capacities which yield integral monotone convergence.