General Probability Measures Research Articles

Intrinsic irreversibility of Kolmogorov dynamical systems

We give the mathematical details and various extensions of the results stated in previous work of Courbage and Prigogine. “Intrinsic random systems” are deterministic and conservative dynamical systems for which we can associate two dissipative Markov processes through a one-to-one “change of representation”, the first leading to equilibrium for t→+∞ and the second for t→-∞. The microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t→+∞. The set of admitted initial conditions D + is then characterized by a non-equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions. Thus, rejected initial states correspond to an infinite amount of information. To realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to Markov processes to such measures. Owing to the non-invariance of D + under the time inversion, the evolution of these states in the new representation can only be given by one of the two possible Markov processes.

Abstract alphabet distortion-rate functions

Two definitions have been given for the distortion-rate function of a sourceuser pair—one involving test channel induced pair probability measures and the other involving general pair probability measures. It is established that both definitions are equivalent for all source-user pairs. Examples are given which exhibit some kinds of the possible pathological behavior of the distortionrate function for general source-user pairs.

Probability measures on fuzzy events in phase space

The notion of fuzzy sample point is introduced, and generalized probability measures on fuzzy events are defined. This leads to the concept of spectral measure on fuzzy events. It is shown that such measures can be associated with quantum-mechanical states when the fuzzy elementary events are represented by Gaussian distributions on phase space.

Measurement in quantum mechanics as a stochastic process on spaces of fuzzy events

The measurement of one or more observables can be considered to yield sample points which are in general fuzzy sets. Operationally these fuzzy sample points are the outcomes of calibration procedures undertaken to ensure the internal consistency of a scheme of measurement. By introducing generalized probability measures on σ-semifields of fuzzy events, one can view a quantum mechanical state as an ensemble of probability measures which specify the likelihood of occurrence of any specific fuzzy sample point at some instant. These sample points are the possible outcomes of any infinitely rapid succession of measurements at that instant of any sequence of observables of the system.

Operational Statistics. I. Basic Concepts

A generalized notion of a ``sample space'' is developed which allows for the simultaneous representation of the outcomes of a set of related ``random experiments.'' Affiliated with each such generalized sample space is a so-called ``logic,'' the elements of which are propositions that can be confirmed or refuted by observing the outcomes of the random experiments. Stochastic models for the experimental situation represented by a given sample space are introduced, and it is shown that such stochastic models induce generalized probability measures on the logic of this sample space.