Product Pre-Measure

In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.EndouGifu Noboru - Gifu National College of Technology Gifu, JapanGrzegorz Bancerek. Towards the construction of a model of Mizar concepts. Formalized Mathematics, 16(2):207-230, 2008. doi:10.2478/v10037-008-0027-x.Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537-541, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc.Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Construction of measure from semialgebra of sets. Formalized Mathematics, 23(4):309-323, 2015. doi:10.1515/forma-2015-0025.Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525-529, 2001.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Hopf extension theorem of measure. Formalized Mathematics, 17(2):157-162, 2009. doi:10.2478/v10037-009-0018-6.Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.P. R. Halmos. Measure Theory. Springer-Verlag, 1974.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3

Possible Experience: from Boole to Bell

Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables.Comment: Published in: EPL, 87 (2009) 6000

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of $p$-integrable functions $L^p$, rely on quantification over proper classes (from the constructive point of view). In this paper we introduce the notions of a pre-measure and pre-integration space, a predicative variation of the Bishop-Cheng notion of a measure space and of an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set $X$ with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces the pre-integration space of simple functions associated to it. Finally, a predicative construction of the canonically integrable functions $L^1$, as the completion of an integration space, is included.Comment: 29 pages; shortened and corrected versio

On the equality of Hausdorff measure and Hausdorff content

We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph-directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. For example, it fails in general for self-conformal sets, self-affine sets and Julia sets. We also give applications of our results concerning Ahlfors regularity. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and $\delta$-approximate packing pre-measure coincide for sufficiently small $\delta>0$.Comment: 21 pages. This version includes applications concerning the weak separation property and Ahlfors regularity. To appear in Journal of Fractal Geometr

On extending the Quantum Measure

We point out that a quantum system with a strongly positive quantum measure or decoherence functional gives rise to a vector valued measure whose domain is the algebra of events or physical questions. This gives an immediate handle on the question of the extension of the decoherence functional to the sigma algebra generated by this algebra of events. It is on the latter that the physical transition amplitudes directly give the decoherence functional. Since the full sigma algebra contains physically interesting questions, like the return question, extending the decoherence functional to these more general questions is important. We show that the decoherence functional, and hence the quantum measure, extends if and only if the associated vector measure does. We give two examples of quantum systems whose decoherence functionals do not extend: one is a unitary system with finitely many states, and the other is a quantum sequential growth model for causal sets. These examples fail to extend in the formal mathematical sense and we speculate on whether the conditions for extension are unphysically strong.Comment: 23 pages, 2 figure

Classifying Cantor Sets by their Fractal Dimensions

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.Comment: 10 pages, revised version. To appear in Proceedings of the AMS

Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures