Summary: For 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 \(\mu \), 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 \(\sigma \)-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 \(\sigma \)-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, \( \mathbb{R}\) is proved to generate this variety, and a concrete model of the free Dedekind \(\sigma \)-complete Riesz spaces with weak unit is exhibited.