A "Bochner-type" integral for Riesz space-valued functions with respect to Riesz space-valued finitely additive measure is introduced, in such a way every bounded function is integrable, with respect to order convergence. Note that is this kind of convergence does not have necessarily a topological character, and we show the substantial differences between convergence in measure with respect to order convergence and the corresponding concept with respect to probability convergence in the space L^0 of all measurable functions. We prove some Vitali and Scheffe'-type theorems, and we show that in general convergence in L^1 does not imply convergence in measure.
Abstract integration in Riesz spaces
BOCCUTO, Antonio
1995
Abstract
A "Bochner-type" integral for Riesz space-valued functions with respect to Riesz space-valued finitely additive measure is introduced, in such a way every bounded function is integrable, with respect to order convergence. Note that is this kind of convergence does not have necessarily a topological character, and we show the substantial differences between convergence in measure with respect to order convergence and the corresponding concept with respect to probability convergence in the space L^0 of all measurable functions. We prove some Vitali and Scheffe'-type theorems, and we show that in general convergence in L^1 does not imply convergence in measure.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
