Borel right process


In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let be a locally compact, separable, metric space.
We denote by the Borel subsets of.
Let be the space of right continuous maps from to that have left limits in,
and for each, denote by the coordinate map at ; for
each, is the value of at.
We denote the universal completion of by.
For each, let
and then, let
For each Borel measurable function on, define, for each,
Since and the mapping given by is right continuous, we see that
for any uniformly continuous function, we have the mapping given by is right continuous.
Therefore, together with the monotone class theorem, for any universally measurable function, the mapping given by, is jointly measurable, that is, measurable, and subsequently, the mapping is also -measurable for all finite measures on and on.
Here,
is the completion of
with respect
to the product measure.
Thus, for any bounded universally measurable function on,
the mapping is Lebesgue measurable, and hence,
for each, one can define
There is enough joint measurability to check that is a Markov resolvent on,
which uniquely associated with the Markovian semigroup.
Consequently, one may apply Fubini's theorem to see that
The following are the defining properties of Borel right processes: