In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
be a locally compact, separable, metric space.
We denote by
the Borel subsets of
be the space of right continuous maps from
that have left limits in
, denote by
the coordinate map at
ω ∈
ω
We denote the universal completion of
, let and then, let For each Borel measurable function
, define, for each
is right continuous, we see that for any uniformly continuous function
Therefore, together with the monotone class theorem, for any universally measurable function
, is jointly measurable, that is,
measurable, and subsequently, the mapping is also
λ ⊗ μ
-measurable for all finite measures
μ
λ ⊗ μ
is the completion of
with respect to the product measure
λ ⊗ μ
Thus, for any bounded universally measurable function
is Lebeague 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:[1]