Editing Mixing (mathematics) (section)

=== Covering families ===
Given a topological space, such as the unit interval (whether it has its end points or not), we can construct a measure on it by taking the open sets, then take their unions, complements, unions, complements, and so on to [[First uncountable ordinal|infinity]], to obtain all the [[Borel set]]s. Next, we define a measure <math>\mu </math> on the Borel sets, then add in all the subsets of measure-zero ("negligible sets"). This is how we obtain the [[Lebesgue measure]] and the Lebesgue measurable sets.

In most applications of ergodic theory, the underlying space is almost-everywhere isomorphic to an open subset of some <math>\R^n</math>, and so it is a Lebesgue measure space. Verifying strong-mixing can be simplified if we only need to check a smaller set of measurable sets.

A covering family <math>\mathcal C</math> is a set of measurable sets, such that any open set is a ''disjoint'' union of sets in it. Compare this with [[Base (topology)|base in topology]], which is less restrictive as it allows non-disjoint unions.

'''Theorem.''' For Lebesgue measure spaces, if <math>T</math> is measure-preserving, and <math>\lim_n \mu(T^{-n}(A)\cap B) = \mu(A)\mu (B)</math> for all <math>A, B</math> in a covering family, then <math>T</math> is strong mixing.

'''Proof.''' Extend the mixing equation from all <math>A, B</math> in the covering family, to all open sets by disjoint union, to all closed sets by taking the complement, to all measurable sets by using the regularity of Lebesgue measure to approximate any set with open and closed sets. Thus, <math>\lim_n \mu(T^{-n}(A)\cap B) = \mu(A)\mu (B)</math> for all measurable {{tmath|1= A, B }}.