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In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let E {\displaystyle E} be a locally compact, separable, metric space.We denote by E {\displaystyle {\mathcal {E}}} the Borel subsets of E {\displaystyle E}.Let Ω {\displaystyle \Omega } be the space of right continuous maps from {\displaystyle } to E {\displaystyle E} that have left limits in E {\displaystyle E} ,and for each t ∈ {\displaystyle t\in } , denote by X t {\displaystyle X_{t}} the coordinate map at t {\displaystyle t} ; foreach ω ∈ Ω {\displaystyle \omega \in \Omega } , X t ∈ E {\displaystyle X_{t}\in E} is the value of ω {\displaystyle \omega } at t {\displaystyle t}. We denote the universal completion of E {\displaystyle {\mathcal {E}}} by E ∗ {\displaystyle {\mathcal {E}}^{*}}.For each t ∈ {\displaystyle t\in } , let
and then, let
For each Borel measurable function f {\displaystyle f} on E {\displaystyle E} , define, for each x ∈ E {\displaystyle x\in E} ,