# Notation and Preliminaries

• R. K. Getoor
Part of the Probability and Its Applications book series (PA)

## Abstract

We shall assume once and for all that
$$X = (\Omega ,\mathcal{F},{{\mathcal{F}}_{t}},{{X}_{t}},{{\theta }_{t}},{{P}^{x}})$$
is a right Markov process as defined in §8 of [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) = △, θ0ω = ω, and θ ω = [△] for all ω ∈ Ω.