Abstract For any real number p ∈ [ 1 , + ∞ ) {p\in[1,+\infty)} , we characterise the operations ℝ I → ℝ {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set ℒ p ⁢ ( μ ) {\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 ℝ {\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, ℝ {\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.

中文翻译：

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保持可积性和截断 Riesz 空间的操作

摘要 对于任何实数 p ∈ [ 1 , + ∞ ) {p\in[1,+\infty)} ，我们刻画操作 ℝ I → ℝ {\mathbb{R}^{I}\to\mathbb{R }} 保持 p 可积性，即对于每个度量 μ，集合 ℒ p ⁢ ( μ ) {\mathcal{L}^{p}(\mu)} 是闭合的操作。我们研究了无穷多的代数，它们的运算正是这样的函数。事实证明，这种多样性与 Dedekind σ-完全截断 Riesz 空间的范畴相吻合，其中截断是 R. N. Ball 意义上的意思。我们还证明了 ℝ {\mathbb{R}} 产生了这种多样性。由此，我们展示了自由 Dedekind σ-完全截断 Riesz 空间的具体模型。对于在有限度量空间上保持 p 可积性的操作，可以获得类似的结果：