Editing Ergodic theory (section)

==Ergodic transformations==
{{Main|Ergodicity}}
Ergodic theory is often concerned with '''ergodic transformations'''.  The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not allow the syrup to remain in a local subregion of the oatmeal, but will distribute the syrup evenly throughout.  At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density.

The formal definition is as follows: 

Let {{math|''T'' : ''X'' &rarr; ''X''}} be a [[measure-preserving transformation]] on a [[measure space]] {{math|(''X'', ''Σ'', ''μ'')}}, with {{math|''μ''(''X'') {{=}} 1}}.  Then {{mvar|T}} is '''ergodic''' if for every {{mvar|E}} in {{mvar|Σ}} with {{math|μ(''T''<sup>−1</sup>(''E'') Δ ''E'') {{=}} 0}} (that is, {{mvar|E}} is [[invariant sigma-algebra|invariant]]), either {{math|''μ''(''E'') {{=}} 0}} or {{math|''μ''(''E'') {{=}} 1}}.

The operator Δ here is the symmetric difference of sets, equivalent to the [[exclusive-or]] operation with respect to set membership. The condition that the symmetric difference be measure zero is called being '''essentially invariant'''.