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Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces

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Abstract

We study non-linear functionals, including quasi-linear functionals, p-conic quasi-linear functionals, d-functionals, r-functionals, and their relationships to deficient topological measures and topological measures on locally compact spaces. We prove representation theorems and show, in particular, that there is an order-preserving, conic-linear bijection between the class of finite deficient topological measures and the class of bounded p-conic quasi-linear functionals. Our results imply known representation theorems for finite topological measures and deficient topological measures. When the space is compact we obtain four equivalent definitions of a quasi-linear functional and four equivalent definitions of functionals corresponding to deficient topological measures.

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Acknowledgements

This work was conducted at the Department of Mathematics at the University of California Santa Barbara. The author would like to thank the department for its hospitality and supportive environment. The author would also like to thank the anonymous reviewers for their valuable comments.

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  1. Department of Mathematics, University of California Santa Barbara, 552 University Rd, Isla Vista, Santa Barbara, CA, 93117, USA

    Svetlana V. Butler

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  1. Svetlana V. Butler
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Correspondence to Svetlana V. Butler.

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Communicated by Manuel Maestre.

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Butler, S.V. Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces. Banach J. Math. Anal. 14, 674–706 (2020). https://doi.org/10.1007/s43037-019-00034-0

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  • DOI: https://doi.org/10.1007/s43037-019-00034-0

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