Fatou's Lemma for Unbounded Gelfand Integrable Mappings

Bernard Cornet (bernard.cornet@univ-paris1.fr) and V. Filipe Martins-da-Rocha
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Bernard Cornet: Department of economics, University of Kansas - KU - University of Kansas [Lawrence]

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Abstract: The objective of this paper is to provide Fatou-type results for sequences of Gelfand integrable mappings defined on a measure space with values in the topological dual of a separable Banach space E. We introduce the assumption of mean weak boundedness that encompasses stronger conditions considered previously. The proof of our Fatou-type results consists in applying successively the scalar version of Komlós' theorem. Since Schmeidler (1970), Fatou's lemma is one of the essential tools in proving the existence of Walrasian equilibria in economic models with a measure space of economic agents as initiated by Aumann (1966). With infinitely many commodities, with commodity space being the topological dual of a separable Banach space, the version of Fatou's lemma with the Gelfand integral applies directly to models of spacial economies (Cornet and Médecin 2000) and models with differentiated commodities (Ostroy and Zame 1994 and Martins-da-Rocha 2003).

Keywords: Fatou's lemma; Banach space; Dual space; Gelfand integral; Komlós limit; Economic equilibrium (search for similar items in EconPapers)
Date: 2021-09
Note: View the original document on HAL open archive server: https://hal.science/hal-03506933v1
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Published in Pure and Applied Functional Analysis, 2021

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