Operations that preserve integrability, and truncated Riesz spaces
(2020) Forum mathematicum — Vol. 32, n° 6, p. 1487-1513 (2020)
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- Abstract
- For any real number p ∈ [ 1, + ∞), we characterise the operations RI →R that preserve p-integrability, i.e., the operations under which, for every measure μ, the set L p (μ) 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 σ-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 σ-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 σ-complete Riesz spaces with weak unit, R is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited.
- Affiliations
University of Milan
Citations
Abbadini, M. (2020). Operations that preserve integrability, and truncated Riesz spaces. Forum mathematicum, 32(6), 1487-1513. https://doi.org/10.1515/forum-2018-0244 (Original work published 2020)