THE BIRKHOFF INTEGRAL

JOSÉ RODRÍGUEZ

There are several extensions of Lebesgue’s theory of integration to the case of functions with values in Banach spaces, being widely known those due to Bochner and Pettis. Birkhoff integral lies strictly between the Bochner and Pettis integrals and has caught the attention of some authors pretty recently. This talk will be devoted to show the new results about this topic that appear in our papers: “Birkhoff integral and the property of Bourgain” (with B. Cascales) and “On the existence of Pettis integrable functions which are not Birkhoff integrable”.

We prove that there is a close relationship between Birkhoff integrability and Bourgain property: a bounded function f defined on a complete probability space with values in a real Banach space X is Birkhoff integrable if, and only if, the set of compositions of f with elements of the dual unit ball has Bourgain property. This fact allows us to characterize the weak Radon-Nikodým property in dual Banach spaces via Birkhoff integrable Radon-Nikodým derivatives. As a consequence we give a solution, in the case of dual Banach spaces, to a problem posed by Fremlin concerning the representation of vector measures as indefinite Pettis integrals of generalized McShane integrable functions.

It is well known that when the range Banach space is separable, then Birkhoff and Pettis integrability coincide. We try to go a bit further when studying the differences between Birkhoff and Pettis integrability in non-separable Banach spaces. We prove that for a weakly Lindelöf determined Banach space X with density character greater than or equal to the continuum, there exists a bounded Pettis integrable function f : [0, 1] ⟶ X which is not Birkhoff integrable. A particular case of our constructions shows that the analogue of Lebesgue’s dominated convergence theorem for the Birkhoff integral does not hold in general.

Departamento de Matemáticas, Universidad de Murcia.

E-mail address: joserr@um.es