Standardness as an Invariant Formulation of Independence

The notion of a homogeneous standard filtration of $\sigma$-algebras was introduced by the author in 1970. The main theorem asserted that a homogeneous filtration is standard, i.e., generated by a sequence of independent random variables (is Bernoulli), if and only if a standardness criterion is satisfied. The author has recently generalized the notion of standardness to arbitrary filtrations. In this paper we give detailed definitions and characterizations of Markov standard filtrations. The notion of standardness is essential for applications of probabilistic, combinatorial, and algebraic nature. At the end of the paper we present new notions related to nonstandard filtrations.