States of classical statistical mechanical systems of infinitely many particles. I

We study a general mathematical model of a classical system of infinitely many point particles. The space X of infinite particle configurations is equipped with a natural topology as well as a measurable structure related to it. It is also connected with a family {X _A } of local spaces of finite configurations indexed by bounded open sets A in the one-particle space E. A theorem analogous to Kolmogoroff's fundamental theorem for stochastic processes is proved, according to which a consistent family { _A } of local probability measures _A defined on the X _A gives rise to a unique probability measure on X. We also study the problem of integral representation for positive linear forms defined over some linear space of real functions on X. We prove that a positive linear form F(f), defined for functions f in the class C+P, admits a uniquely determined integral representation F(f)= f () d, where is a probability measure over X.