Measures from infinite-dimensional gauge integration

Cyril Levy (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 INUC - Institut national universitaire Champollion [Albi]

We construct and investigate an integration process for infinite products of compact metrizable spaces that generalizes the standard Henstock-Kurzweil gauge integral. The integral we define here relies on gauge functions that are valued in the set of divisions of the space. We show in particular that this integration theory provides a unified setting for the study of non-absolute infinite-dimensional integrals such as the gauge integrals on T of Muldowney and the construction of several type of measures, such as the Lebesgue measure, the Gaussian measures on Hilbert spaces, the Wiener measure, or the Haar measure on the infinite dimensional torus. Furthermore, we characterize Lebesgue-integrable functions for those measures as measurable absolutely gauge inte-grable functions.

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Cyril Levy. Measures from infinite-dimensional gauge integration. 2019. ⟨hal-02308813⟩