A σ-linked boolean algebra without positive measure

σ-ring and σ-algebra of Sets1

In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets here to avoid such a confusion. Ring ...

Noise as a Boolean Algebra of Σ -fields by Boris Tsirelson

Introduction. The product of two measure spaces, widely known among mathematicians, leads to the tensor product of the corresponding Hilbert spaces L2. The less widely known product of an infinite sequence of probability spaces leads to the so-called infinite tensor product space. A continuous product of probability spaces, used in the theory of noises, leads to a continuous tensor product of H...

A Bernoulli Toral Linked Twist Map without Positive Lyapunov Exponents

The presence of positive Lyapunov exponents in a dynamical system is often taken to be equivalent to the chaotic behavior of that system. We construct a Bernoulli toral linked twist map which has positive Lyapunov exponents and local stable and unstable manifolds defined only on a set of measure zero. This is a deterministic dynamical system with the strongest stochastic property, yet it has po...

Ternary Boolean Algebra

Boolean Algebra Approximations

Knight and Stob proved that every low4 Boolean algebra is 0-isomorphic to a computable one. Furthermore, for n = 1, 2, 3, 4, every lown Boolean algebra is 0-isomorphic to a computable one. We show that this is not true for n = 5: there is a low5 Boolean algebra that is not 0-isomorphic to any computable Boolean algebra. It is worth remarking that, because of the machinery developed, the proof u...