Measure, Probability and Function Spaces

A probability distribution is assumed to be characterized by a measure on the state space of the problem which is the integral of a probability density function deflned on the phase space of the problem that assigns nonnegative real numbers to each state. We outline the technical setting here. To gain a fuller understanding of the mathematical issues involved in this chapter, it may be helpful to refer to an introductory text on analysis such as [26,317]. 

Assume, as usual, a system of N atoms with phase space having dimension m (typically m = 2Nc = 2Nd = 6N). Let D c be a suitable set comprising the collection of admissible phase space points z. A density on D is a function p D R such that (i) p(z) 0, z e D, and (ii) pdo) oo, where doj = dzidz2 dZm (i.e. pdco is the integral of the measurable function p in the Lebesgue sense). A probability density function (or p.d.f.) on D is a density which is normalized so that 

Any density (with positive total integral) can be normalized by dividing by pdco. For any measurable subset A of K, and probability density p, we define 

It is automatic that 0 Pr(yi) 1. Certain properties need to be assumed regarding p and the way that this is formalized in mathematics is by description of the vector space of functions to which p belongs. 

A useful class of functions from a domain D c R to R are those that satisfy the following boundedness condition  

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