A categorical derivation of Lebesgue integration

2020/11/01 by Tom Leinster, Leinster, Tom · 2 citations
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Category Theory (math.CT) #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2011.00412

openalex publication_date 2020/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We identify simple universal properties that uniquely characterize the Lebesgue L^p spaces. There are two main theorems. The first states that the Banach space L^p[0, 1], equipped with a small amount of extra structure, is initial as such. The second states that the L^p functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces ℓ^p and c₀, as well as the functor L² taking values in Hilbert spaces.

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• Orthocomplemented subspaces and partial projections on a Hilbert space
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