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We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures.

We prove the Bochner-Kolmogorov theorem on the existence of the limit of projective systems of second countable Hausdorff (non-metrizable) spaces with tight probabilities, such that the projection mappings are merely measurable functions. Our direct and transparent approach (using Lusin’s theorem) should be compared with the previous work where the spaces are assumed metrizable and the main idea was to reduce the general context to a regular one via some isomorphisms. The motivation of the revisit of this classical result is an application to the construction of the continuous time fragmentation processes and related branching processes, based on a measurable identification between the space of all fragmentation sizes considered by J. Bertoin and the limit of a projective system of spaces of finite configurations.