A general Fubini theorem for the Riesz paradigm

TL;DR

This paper establishes a generalized Fubini theorem within the framework of monoidal and enriched category theory, extending classical results to convergence spaces and continuous linear functionals.

Contribution

It introduces an abstract Fubini theorem in category theory and derives a new Fubini theorem for integrals on convergence spaces, unifying and generalizing classical measure theory results.

Findings

Proves a monad of natural distributions is commutative under certain conditions.

Derives a Fubini theorem for continuous linear functionals on convergence spaces.

Generalizes classical Fubini theorem for Radon measures to a categorical setting.

Abstract

We prove an abstract Fubini-type theorem in the context of monoidal and enriched category theory, and as a corollary we establish a Fubini theorem for integrals on arbitrary convergence spaces that generalizes (and entails) the classical Fubini theorem for Radon measures on compact Hausdorff spaces. Given a symmetric monoidal closed adjunction satisfying certain hypotheses, we show that an associated monad of natural distributions D is commutative. Applying this result to the monoidal adjunction between convergence spaces and convergence vector spaces, the commutativity of D amounts to a Fubini theorem for continuous linear functionals on the space of scalar functions on an arbitrary convergence space.