Lower semicontinuity of monotone functionals in the mixed topology on C b

Max Nendel
Additional contact information
Max Nendel: Center for Mathematical Economics, Bielefeld University

No 723, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

Abstract: The main result of this paper characterizes the continuity from below of monotone functionals on the space $C_b$ of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair $(C_b, ca)$, where $ca$ denotes the space of all countably additive signed Borel measures of finite variation. Hence, lower semicontinuity in the mixed topology of convex monotone maps $C_b \to \mathbb{R}$ is equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g., in the context of risk measures and super hedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. In a second step, the paper provides a characterization of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on $C_b$ taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm bounded sets.

Keywords: Risk measure; monotone functional; Choquet integral; continuity from below; lower semicontinuity; mixed topology; Mackey topology (search for similar items in EconPapers)
Pages: 21
Date: 2025-07-18
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://pub.uni-bielefeld.de/download/3005288/3005289 First Version, 2024 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bie:wpaper:723

Access Statistics for this paper

More papers in Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University Contact information at EDIRC.
Bibliographic data for series maintained by Bettina Weingarten ().