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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.
Admissible vectors lead to frames or coherent states under the action of a group by means of square integrable representations. This work shows that admissible vectors can be seen as weights with central support on the (left) group von Neumann algebr
We clarify the relation between inverse systems, the Radon-Nikodym property, the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in our earlier paper on differentiability of Lipschitz maps into Banach spaces.
If $mu_1,mu_2,dots$ are positive measures on a measurable space $(X,Sigma)$ and $v_1,v_2, dots$ are elements of a Banach space ${mathbb E}$ such that $sum_{n=1}^infty |v_n| mu_n(X) < infty$, then $omega (S)= sum_{n=1}^infty v_n mu_n(S)$ defines a vec
We investigate the effectivizations of several equivalent definitions of quasi-Polish spaces and study which characterizations hold effectively. Being a computable effectively open image of the Baire space is a robust notion that admits several chara
Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only if it has a