Stochastic processes in terms of inner premeasures

Abstract

In a recent paper the author used his work in measure and integration to obtain the projective limit theorem of Kolmogorov in a comprehensive version in terms of inner premeasures. In the present paper the issue is the influence of the new theorem on the notion of stochastic processes. It leads to essential improvements in the foundation of special processes, of the Wiener process in the previous paper and of the Poisson process in the present one. But it also forms the basis for a natural redefinition of the entire notion. The stochastic processes in the reformed sense are in one-to-one correspondence with the traditional ones in case that the state space is a Polish topological space with its Borel σ algebra, but the sizes and procedures are quite different. The present approach makes an old idea of Kakutani come true, but with due adaptations.