Pseudo-Fubini Real-Entire Functions on the Plane

Abstract

In this note, it is proved the existence of a c -dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real C∞ -functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained.