Localisation of continuity to bounded sets for nonmetrisable vector topologies and its applications to economic equilibrium theory

AbstractWe present two ‘localisation’ techniques which facilitate verifications of the topological properties of sets, functions and correspondences needed in, e.g., economic equilibrium analysis with infinite-dimensional commodity spaces. The first uses the Krein-Smulian theorem, which shows that weak∗ upper semicontinuity of a concave function on a dual Banach space is equivalent to bounded weak∗ u.s. continuity. The second is based on the continuity of lattice operations: for a nondecreasing function on a topological vector lattice, we show that lower semicontinuity on a set bounded from below is equivalent to l.s. continuity on bounded subsets. In the case of L∞, we use convergence in measure to establish thatsequential semicontinuity, lower for the Mackey topology or upper for the weak∗ one, is equivalent to semicontinuity. This greatly simplifies some arguments; e.g., the Mackey continuity of a concave, nondecreasing integral functional on L+∞ becomes an immediate consequence of Lebesgue's Dominated Convergence Theorem. Other uses in mathematical economics are also discussed