### Abstract:

The work in this thesis falls into two parts. The first part
presents a rigorous theory of probability densities and correlation
functions within the framework of the exponential spaces of D. S.
Carter. The second part extends this discussion to include infinite
systems.
The first part begins by considering an algebra of complex-valued
functions defined on the exponential of a set. Multiplication
in this algebra corresponds to the star product used by Schwartz
and Ruelle. A functional calculus is defined on this algebra, which
provides for the discussion of Ursell functions. By introducing a
special subspace of integrable functions, a systematic theory is
obtained, in which probability densities and correlation functions
belonging to this-subspace are related by a pair of mutually inverse
integral operators. The second part deals with various ways of representing the
statistical state of an infinite system. The existence of states of an
infinite system is established as limits of a special class of states
of finite systems. The equilibrium states considered by Ruelle are
shown to be included in this class.