7 research outputs found
A Computable Version of the Daniell-Stone Theorem on Integration and Linear Functionals
AbstractFor every measure ÎŒ, the integral I:fâŠâ«fdÎŒ is a linear functional on the set of real measurable functions. By the Daniell-Stone theorem, for every abstract integral Î:FâR on a stone vector lattice F of real functions f:ΩâR there is a measure ÎŒ such that â«fdÎŒ=Î(f) for all fâF. In this paper we prove a computable version of this theorem
Computability of the Radon-Nikodym derivative
We study the computational content of the Radon-Nokodym theorem from measure
theory in the framework of the representation approach to computable analysis.
We define computable measurable spaces and canonical representations of the
measures and the integrable functions on such spaces. For functions f,g on
represented sets, f is W-reducible to g if f can be computed by applying the
function g at most once. Let RN be the Radon-Nikodym operator on the space
under consideration and let EC be the non-computable operator mapping every
enumeration of a set of natural numbers to its characteristic function. We
prove that for every computable measurable space, RN is W-reducible to EC, and
we construct a computable measurable space for which EC is W-reducible to RN
Representations of measurable sets in computable measure theory
This article is a fundamental study in computable measure theory. We use the
framework of TTE, the representation approach, where computability on an
abstract set X is defined by representing its elements with concrete "names",
possibly countably infinite, over some alphabet {\Sigma}. As a basic
computability structure we consider a computable measure on a computable
-algebra. We introduce and compare w.r.t. reducibility several natural
representations of measurable sets. They are admissible and generally form four
different equivalence classes. We then compare our representations with those
introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our
representations is the most useful one for studying computability on measurable
functions
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Computable Measure Theory and Algorithmic Randomness
International audienceWe provide a survey of recent results in computable measure and probability theory, from both the perspectives of computable analysis and algorithmic randomness, and discuss the relations between them