Projective limits via inner premeasures and the true Wiener measure

The paper continues the author’s work in measure and integration, which is an attempt at unified systematization. It establishes projective limit theorems of the Prokhorov and Kolmogorov types in terms of inner premeasures. Then it specializes to obtain the (one-dimensional) Wiener measure on the space of real-valued functions on the positive halfline as a probability measure defined on an immense domain: In particular the subspace of continuous functions will be measurable of full measure and not merely of full outer measure, as the usual projective limit theorems permit to conclude. The present paper wants to continue the author’s chain of contributions to measure and integration. This is an attempt at unified systematization, with the particular aim to incorporate the topological theory into the abstract one. The basic idea is to develop and to convert the classical extension method due to Carathéodory into a few different procedures. These procedures are parallel to each other, but diversified in two respects: On the one hand as to their basic inner or outer character, and on the other hand as to their discrete, sequential or nonsequential limit behaviour. Since 1996 there are the book [11] (cited as MI) and a series of subsequent papers, and the recent survey article [15]. A number of topics has been treated with unified results which extend and improve the former ones in both of the conventional theories. A typical example is the formation of products in MI chapter VII and [13]. The present paper will be devoted to the formation of projective limits, like the formation of products in the inner context. This is a topic of particular importance, and in fact considered to be a crucial one. We quote a statement from Dellacherie-Meyer [4] of 1978 pp.65/66. If abstract measure theory ... is compared to the theory of Radon measures ..., it may seem that the latter is superior to 1991 Mathematics Subject Classification. 28A12, 28A35, 28C20, 60A10.