The aim of the present paper, which will be published in several parts, is to study MARKOV processes, their semigroups of transition probabilities, and resolvents under purely probabilistic assumptions in arbitrary measurable spaces. The hypotheses that will be used are natural and will be satisfied in most applications. Our approach is based on the systematical use of r-excessive and r-supermedian functions.
The main purpose of the paper is the elimination of topological assumptions in the theory of MARKOV processes. This means that we do not use any topology, except for the topology of the real numbers and, possibly, topologies in the state space which are, however, related with the given MARKOV process in a natural way. In the state space of a MARKOV process various topologies of fundamental importance can be introduced: RAY topologies, the fine topology, the topology of stochastic continuity.
E. B. DYNKIN [5]
also considered the problem of eliminating topological assumptions but his methods and results are quite different.
For every semigroup of transition probabilities (resp. substochastic resolvent) we shall introduce and study the topology of (resp. essentially) stochastic continuity. This notion leads us to the definition of a standard semigroq (resp. resolvent). Standard semigroups (resp. resolvents) have a number of desirable properties.
I n sortie applications, so-called branching points appear. Thus we inust consider a, richer class of seniigroups which will be called right continuous semigroup. For later investigations of regularity properties at the left of MARKOV processes we also introduce left continuous semigroups and collect some important propperties of them.
Using these semigroup methods we next come to the study of the sample path6 of i n A R ~~~ processes. The notions introduced in the first parts allow us to weaken the "HYPOTH~SES DROITES" of P. A. MEYER and J. B. \'ALSH [15]. In fact, the first of these hypothescs, concerning with the existence of a MARKOV process having right continuous sample paths (the &,ate space is assumed to be a BOREL set in a compact metric space), will be completely eliminated. For )* -70 Engelbert, Markov Processes in General State Spaces MARKOV Imjcesaes satisfying the HY P O T ~S E S DROITES also see the recently 1)ublished detailed representation of R. K. GETOOR [7].
On this basis, an itxiotriatical treatment is given of ~I A R K O V processes with regular sample paths in arbitrary measurable spaces, including processes sibtisfyitlg the HYPOTH~SES DROITES. A further hypothesis is restricting this clilss of MARROV processes to a class of processes with regularity properties at the left.
Tho axiomaticitl itpproitch presented in the paper is justified by several consfruclion theorem. Using compactification methods of D. RAY [ 121 and k ' . KNIGHT [8], necessary and sufficient conditions are given for constructing strong X~AKKOV processes w i t h values in a, posuibly, greater state space.
Finally, some applications to the genertd theory of MARKOV processes are give t i .
Similar questions were previously considered by many authors. Except for those papers cited above we only recall the papers of KUNITA ancl \'ATANABE [ O ] , P. A. MEYER [ 111 for locally compact state spaces with countable base, J. L. DOOB [3] for st.andard semigroups on the discrete state space and C . T. SHIH "41. The most iml)ortant results of the present paper were nnnouiiced i n [(;I. 1. :Preliminaries Semigrovps lxnd Resolvenfs 1. Let (E, 8 ) be a measurable space. By B = B ( E , 8 ) ; m i B , = B + ( E , 8 ) we denote tho collwtions of bounded irieitsurable funct.ioiis and nonnept ive itieasurable f'unctioils respectively.1.) Clearly, B is i.i B.4NACII s1)iice with reslbect t o the iiorni 11 .I/ clefinetl hy llfll = s u p If(z)l. In the following, uniforni convergence, uniform c:losure etc. always iiiean the convergence of functions, the dosure of ii set of functions etc. in t h i s BANACH spa~e.