Projective limits via inner premeasures and the true Wiener measure

Abstract

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.KeywordsProjective limit theorems of the Prokhorov and Kolmogorov typesProkhorov conditioninner premeasures and their inner extensionsdirect and inverse images of inner premeasurestransplantation theoremsWiener measure and Wiener premeasureBrownian convolution semigroup