AbstractFor any real number {p\in[1,+\infty)}, we characterise the operations {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set {\mathcal{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 σ-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball.
We also prove that {\mathbb{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, {\mathbb{R}} is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited.