On the continuity of the concave integral

Whenever a functional is concave it is natural to ask whether its sendograph is a closed convex set. If so, the Hahn–Banach theory implies that the functional can be represented as the infimum of all continuous linear functionals greater than or equal to it. We refer to such representation as a dual representation. Dominated convergence of the concave integral for capacities is characterized in terms of dual representation whenever sequences of functions converge pointwise outside a set of zero capacity. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. e-mail: [email protected] The author would like to thank two anonymous referees and editor of Fuzzy Sets and Systems, and especially Ehud Lehrer for his continued guidance. 1