A stochastic process is defined to be a function that maps the set to random variable .
Let and be the values of the random variables and B be its Borel field. State space is denoted by
Let be the -field of some set , it then must follow
E.g. then an example of the sigma-algebra generated from is .
Borel Field or Borel-sigma-algebra generated from is defined as the sigma-algebra of all open subsets of .
defines a probability space
Probability measure P is a function which associates a number P(A) to each set with the following properties:
- Associative properties on disjoint sets , that is
Note: See as the probability that the events occurs.