Notions of Probabilistic Computability on Represented Spaces

Abstract: we define and compare several probabilistic notions of computability for mappings from represented spaces (that are equipped with a measure or outer mea- sure) into computable metric spaces. we thereby generalize definitions by [ko 1991] and parker (see [parker 2003, parker 2005, parker 2006]), and furthermore introduce the new notion of computability in the mean. some results employ a notion of com- putable measure that originates in definitions by [weihrauch 1999] and [schr¨oder 2007]. in the spirit of the well-known representation theorem (see [weihrauch 2000]), we establish dependencies between the probabilistic computability notions and classical properties of mappings. we furthermore present various results on the computability of vector-valued integration, composition of mappings, and images of measures. finally, we discuss certain measurability issues arising in connection with our definitions.