Measure Theory Exercises (Kolmogorov Consistency Theorem)

(b) Show, conversely. (using Ex. 4.4.5) that if the system of transition proba bilities' satisfies the Chapman-Kolmogorov equations, then there exists a unique probability measure P defined on (S , B) = (R10.30), BE10. x, By) satisfying conditions (i) and (ii) of (a) above. EX. 4.4.7 If { P, : t = [0, x )} is any family of probability measures on (R . B.) such that ( > P.(E) is a measurable map for each Borel set E, define Po = P and P (s, x; t, E) = P,(E) and deduce from Ex. 4.4.6 that there exists a unique probability measure P on (2 . B) = (R10.). /E0. x) BR) such that PHWES : W(t) E Al. ... . W(t,) E An)) = Pn(Al) . . . P., (A,) for arbitrary 0