Let be a filtered probability space with a discrete filtration (although the results we are going to discuss hold for continuous random processes as well). We say that a random process on possesses the Markov property if
for all . We see this often in the following form
or simply with with . This is the Markov property.
The following is just wrong: For a sequence of sets where
A very easy and straightforward way to verify that this is false is to set , that is, provide no information about and provide the information , i.e., which actually offers some information. Then, according to the wrong statement above, it would be
Although, it should naturally be . That is, the information that is completely expunged.
Reference: K.L. Chung, Green, Brown and Probability, World Scientific, 1995 [Chap. 5].