ABSTRACT
This chapter is devoted to applications of conditioning to two of
the most important areas of Probability Theory, namely martingales
and Markov processes. The former uses the general properties of condi-
tional expectations while the latter depends essentially on the regularity
of conditional measures. Here basic results on the mean and pointwise
convergence of (directed indexed) martingale limit theorems, as well
as structural properties of Markov processes are presented. These in-
clude the existence and continuity properties of Markov processes under
different conditions. However, the problems on the evaluation of con-
ditional expectations of functionals on these processes is used with an
“idealistic approach” (Bishop’s terminology) although the basic con-
structive problem still awaits a satisfactory solution.
