Construction

Let be a measurable space. Suppose , then we the cartesian of by

is called an cylinder set if for all but finite number of . Then collection of cylinder sets is denoted by

Easy to verify that is a system(algebra). In this section, we will assign a probability value for each cylinder set,

We define

If , we we define

Easy to verity that is consistence and thus there exists a unique such that

On , then canonical process is a Markov process on with respect to the natural filtration.

Note that

where

where for all but .

We need to verity

Equivalently,

We can consider the cylinder sets in , and then apply dynkin's theorem. A cylinder set in is in the form of

Then

For , when , then

It is valid for all bounded measurable function . Since is a bounded measurable function, then