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Journal of Korean institute of intelligent systems

The concept of σ-morphism as a probability measure on the set of effects

Yong Sik Yun, Kyoung-Hun Kang, Jin-Won Park

Published Jun 25, 2009• 1 citation
Publisher

Abstract

이 논문에서는 사건과 확률변수를 각각 일반화한 이펙트와 옵저버블을 소개하였다. 그리고 $\sigma$ -함수의 개념을 소개하고 이펙트 집합위에서의 확률측도로서의 $\sigma$ -함수의 성질을 조사하였다. In this paper, we introduce the concepts of effects and observable as generalizations of event and random variable, respectively. Also, we introduce the concept of $\sigma$ -morphism and we investigate some results on $\sigma$ -morphism as a probability measure on the set of effects.

DOI
10.5391/jkiis.2009.19.3.371

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이 논문에서는 사건과 확률변수를 각각 일반화한 이펙트와 옵저버블을 소개하였다. 그리고 σ\sigmaσ -함수의 개념을 소개하고 이펙트 집합위에서의 확률측도로서의 σ\sigmaσ -함수의 성질을 조사하였다. In this paper, we introduce the concepts of effects and observable as generalizations of event and random variable, respectively. Also, we introduce the concept of σ\sigmaσ -morphism and we investigate some results on σ\sigmaσ -morphism as a probability measure on the set of effects.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B \mathcal{B} B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B \mathcal{B} B X constitute a convex set whose extreme points are degenerate probability measures.
The sum of observables on a σ\sigma σ σ -distributive lattice effect algebra
TEM 이론의 기초가 되는 델타함수, 콘볼루션 적분, 퓨리에 변환에 관한 개념을 소개하고 이에 대한 응용으로 슬릿함수, 현저한 폭을 갖는 2개의 슬릿, 유한 크기의 파동 train, 좁은 슬릿의 주기적인 배열, 임의의 주기 함수, diffraction grating, 회절 격자, 그리고 gaussian 함수에서의 퓨리에 변환에 관한 수학적인 방법의 적용을 소개하였다. 【In this review, the fundamental concepts of delta function, convolution integral and Fourier transformation are discussed. The applications of Fourier transformation to slit function, two very narrow slits, two slits of appreciable width, periodic array of narrow slits, arbitary periodic function, diffraction gratings and gaussian functions are also introduced.】
(i) is therefore a good reflection of the crux of the measurement process. Here H(X) is the which characterizes the a priori of the quantity being measured, and H(X/Y) is the total conditional which characterizes the of the quantity being measured which remains after the measurement result Y has been obtained. Let us examine the application of the concept in the case of continuous distributions. The value of the as a numerical characteristic of a continuous random variable is rejected in [12] on the basis that the equals ~ for continuous random variables. The introduction of the probability density p(x) could be rejected on approximately the same basis by considering that this quantity does not reflect the probabilistic characteristic by not being a probability; and only the product p(x)dx possesses the probabilistic interpretation, but equals 0 for continuous quantities and hence loses meaning. Wherever required and possible, the probability is approximated by the expression p(x)Ax, where &x, albeit small, is a finite quantity. The concepts of reduced (differential) entropy and quantization step have an analogous (but not identical) meaning [16]. If is understood to be the mathematical expectation of the logarithm of the probability of random variable values, then it is indeed impossible to relate this concept to continuous random variables. On the other hand, the initial physical meaning, uncertainty, vanishes in the reduced entropy concept. The initial concept uncertainty not only remains real for continuous random variables in the examination of the quantization step concept, but stands out more, exposing most completely the content of the Heisenberg principle and the negative principle of information [5, 7], according to which no quantity being observed can be measured with arbitrarily high accuracy. The limit is not set on this accuracy by the imperfection of the instruments but by objective laws of nature.
Reichenbach's notion of common cause In what follows ( X, S, p ) denotes a classical Kolmogorovian probability space with Boolean algebra S of subsets of a set X (with respect to the set theoretic operations ∩, ∪ and A ⊥ = X\A as Boolean algebra operations) and with the probability measure p on S . (See the Appendix for a concise review of the basic concepts of measure theoretic probability theory.) Elements of S are called random events (elements of X are sometimes called (random) elementary events). It is common to assume in probability theory that p is a countably additive (also called σ-additive) and not just a finitely additive measure but the assumption of countable additivity is somewhat controversial in the philosophical literature. The distinction between countable and finite additivity will not play any role in Chapters 3–5, 7 and 9: the results presented are valid under the assumption of either finite or countable additivity. Countable additivity will play a role in Chapters 6 and 8, however, where the problem of correlations in nonclassical (quantum) probability spaces will be investigated, and where the quantum counterpart of p will be assumed to be countably additive (“normal” in the terminology of the theory von Neumann algebras). If S has a finite number of elements, then it is the power set P(X) of a set X having n ＜ ∞ elements denoted by a i ( i = 1, 2 … n ); in this case we write S n .