A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the ”extended” real plane. Its interest is motivated also by the fact that often simple or multiple integrals do not exist in the sense of Lebesgue, but only as improper ones, while they can be viewed as Kurzweil-Henstock integrals. Moreover in applications involving stochastic processes and integration, it is advisable to work with mappings taking values in functional spaces, like L^1 or L^0: more generally, we shall consider integration for Riesz space-valued functions. In this paper we prove the following Theorem Let J = H×K be a closed rectangle, f : J -> R be KH-integrable and w-dominated, and suppose that the function Q(x) = int_K f(x, y) dy is well-defined with respect to a common regulator in H \ N, where N has Lebesgue measure zero. Then Q is KH-integrable on H and int_H (int_k f(x, y)dy) dx = \int_(H x K) f(x,y) dx dy. A crucial hypothesis in this theorem is the w-domination of f (see Definition 3.7), related to the absolute KH-integrability of f (see Section 5).
A Fubini Theorem in Riesz Spaces for the Kurzweil-Henstock Integral
BOCCUTO, Antonio;CANDELORO, Domenico;SAMBUCINI, Anna Rita
2011
Abstract
A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the ”extended” real plane. Its interest is motivated also by the fact that often simple or multiple integrals do not exist in the sense of Lebesgue, but only as improper ones, while they can be viewed as Kurzweil-Henstock integrals. Moreover in applications involving stochastic processes and integration, it is advisable to work with mappings taking values in functional spaces, like L^1 or L^0: more generally, we shall consider integration for Riesz space-valued functions. In this paper we prove the following Theorem Let J = H×K be a closed rectangle, f : J -> R be KH-integrable and w-dominated, and suppose that the function Q(x) = int_K f(x, y) dy is well-defined with respect to a common regulator in H \ N, where N has Lebesgue measure zero. Then Q is KH-integrable on H and int_H (int_k f(x, y)dy) dx = \int_(H x K) f(x,y) dx dy. A crucial hypothesis in this theorem is the w-domination of f (see Definition 3.7), related to the absolute KH-integrability of f (see Section 5).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
