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

The induced capacity and Choquet integral monotone convergece