Excessive Measures pp 1-5 | Cite as

# Notation and Preliminaries

Chapter

## Abstract

We shall assume once and for all that is a right Markov process as defined in §8 of [

$$X = (\Omega ,\mathcal{F},{{\mathcal{F}}_{t}},{{X}_{t}},{{\theta }_{t}},{{P}^{x}})$$

**S**] with state space (*E,ε*),semigroup (*P*_{ t }), and resolvent (*U*^{ q }). To be explicit*E*is a separable Radon space and*ε*is the Borel*σ*-lgebra of*E*. A cemetery point △ is adjoined to*E*as an isolated point and*E*_{△}: =*E*∪ {△},*ε*_{△}:*= σ*(*ε*∪ {△}). (The symbol “: =” should be read as “is defined to be”.) We suppose that [**S**, (20.5)] holds; that is,*X*_{ t }(*ω*) = △ implies that*X*_{ s }(*ω*) = △ for all*s ≥ t*and that there is a point [△] in Ω (the dead path) with X_{ t }([△]) = △ for all*t ≥*0. Of course,*ζ*= inf {*t*:*X*_{ t }= △} is the lifetime of*X*. The filtration (*F,F*_{ t }) is the augmented natural filtration of*X*, [**S**, (3.3)]. We shall always use*ε*to denote the Borel*σ*-algebra of*E*in the original topology of*E*. Beginning in §20, Sharpe uses*ε*to denote the Borel*σ*-algebra of*E*in the Ray topology. We shall*not*use this convention. We shall write*ε*^{ r }for the*σ*-algebra of Ray Borel sets. These assumptions on*X*are weaker than those in [**G**] or [**DM**, XVI4]. Beginning in §6 we shall make an additional assumption on*X*. (See (6.2)). To avoid trivialities*we assume throughout this monograph that X*_{∞}(*ω*) = △,*θ*_{0}*ω*=*ω, and θ*_{ ∞ }*ω*= [△]*for all ω*∈ Ω.### Keywords

Filtration Radon## Preview

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© Birkhäuser Boston 1990