Definition
Ergodic transformation
A probability-preserving transformation with no invariant event of intermediate probability.
A probability-preserving transformation is ergodic if
Equivalently, its invariant sigma-algebra is trivial modulo null sets.
Meaning
There is no decomposition into two measurable invariant pieces of positive probability. Ergodicity does not assert a single orbit, nor does it require all orbits to be dense. On an atomless space, every orbit of a single invertible transformation is countable and has measure zero.
Examples
The finite uniform systems that are ergodic are exactly single cycles. An irrational circle rotation is ergodic, while an orthogonal map of a finite-dimensional Gaussian space is not.