For any real number p is an element of [1, +infinity), we characterise the operations RI -> R that preserve p-integrability, i.e., the operations under which, for every measure mu, the set L-p(mu) is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind sigma-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that R generates this variety. From this, we exhibit a concrete model of the free Dedekind sigma-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind sigma-complete Riesz spaces with weak unit, R is proved to generate this variety, and a concrete model of the free Dedekind a-complete Riesz spaces with weak unit is exhibited.

Operations that preserve integrability, and truncated Riesz spaces / M. Abbadini. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - 32:6(2020), pp. 1487-1513. [10.1515/forum-2018-0244]

Operations that preserve integrability, and truncated Riesz spaces

M. Abbadini
2020

Abstract

For any real number p is an element of [1, +infinity), we characterise the operations RI -> R that preserve p-integrability, i.e., the operations under which, for every measure mu, the set L-p(mu) is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind sigma-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that R generates this variety. From this, we exhibit a concrete model of the free Dedekind sigma-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind sigma-complete Riesz spaces with weak unit, R is proved to generate this variety, and a concrete model of the free Dedekind a-complete Riesz spaces with weak unit is exhibited.
Integrable functions; L p; Riesz space; vector lattice; σ-completeness; weak unit; infinitary variety; equational classes; axiomatisation; free algebra; generation
Settore MAT/01 - Logica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/857968
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