The induced capacity and Choquet integral monotone convergece

TL;DR

This paper investigates how the Choquet integral behaves under monotone convergence when using capacities induced by partial probability information, providing characterizations for convergence conditions.

Contribution

It characterizes the properties of sub-$\sigma$-algebras and induced capacities that ensure monotone convergence of the Choquet integral.

Findings

Identifies conditions for monotone convergence of the Choquet integral.

Provides a characterization of capacities induced by partial probability measures.

Connects induced capacities with integrals on sub-algebras.

Abstract

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.