On the continuity of the concave integral

摘要

Whenever a functional is concave it is natural to ask whether its sendoaraph is a closed convex set. If so, the Hahn-Banach theory implies that the functional can be represented as the intimum 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.