Borel measurability in linear algebra

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire sp...

Measurability in Linear and Non-Linear Quantum Mechanical Systems

The measurability by means of continuous measurements, of an observable A(t0), at an instant, and of a time averaged observable, Ā = 1/T ∫ A(t)dt, is examined for linear and in particular for non-linear quantum mechanical systems. We argue that only when the exact (non-perturbative) solution is known, an exact measurement may be possible. A perturbative approach is shown to fail in the non-line...

An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability

A Closed Algebra with a Non-Borel Clone and an Ideal with a Borel Clone

Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest t...

Measurability of linear operators in the Skorokhod topology

It is proved that bounded linear operators on Banach spaces of “cadlag” functions are measurable with respect to the Borel σ-algebra associated with the Skorokhod topology.