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Krivine’s Function Calculus and Bochner Integration

Part of:Topological linear spaces and related structuresNormed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  15 October 2018

V. G. Troitsky and
M. S. Türer
V. G. Troitsky
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada Email: troitsky@ualberta.ca
M. S. Türer
Affiliation:
Department of Mathematics and Computer Science, İstanbul Kültür University, Bakırköy 34156, İstanbul, Turkey Email: m.turer@iku.edu.tr
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Abstract

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We prove that Krivine’s Function Calculus is compatible with integration. Let $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, $X$ a Banach lattice, $\mathbf{x}\in X^{n}$, and $f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that $f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and $f(\mathbf{s},\cdot )$ is integrable for every $\mathbf{s}\in \mathbb{R}^{n}$. Put $F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define $F(\mathbf{x})$ and $f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions $F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.

Keywords

Banach latticefunction calculusBochner integral

MSC classification

Primary:46B42: Banach lattices
Secondary:46A40: Ordered topological linear spaces, vector lattices

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019, pp. 663 - 669
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The first author was supported by an NSERC grant.

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