Ergodicity is the property of a measure-preserving dynamical system that every invariant measurable set
has either zero or full measure. For a measure-preserving
transformation 
of a probability space 
,
the precise condition is
 |
for every 
, where 
denotes the symmetric
difference. Thus the system cannot be decomposed into two invariant measurable
parts of positive measure (Walters 1982, Tao 2008).
An ergodic transformation is a map with this property relative to the specified invariant measure. An ergodic measure is an invariant measure that gives the property for a specified map. Ergodicity therefore concerns the transformation and the measure together.
Every mixing transformation is ergodic. A Bernoulli scheme is mixing and hence ergodic, so the Bernoulli property is a stronger condition than ergodicity (Walters 1982).
