Measure-preserving dynamical system

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory.

It is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

with the following structure:

X is a set,

is a σ-algebra over X,

is a probability measure, so that m(X) = 1, and

is a measurable transformation which preserves the measure m, i. e. each measurable satisfies

m(T ^{- 1}A) = m(A).

For example, m could be the normalised angle measure dθ/2π on the unit circle, and T a rotation.

One may wonder why the seemingly simpler identity

m(T(A)) = m(A)

is not used. Here is the problem: suppose T : [0, 1] → [0, 1] is defined by T(x) = (4x mod 1), i.e., T(x) is the "fractional part" of 4x. Then the interval [0.01, 0.02] is mapped to an interval four times as long as itself, but nonetheless the measure of T ⁻¹( [0.04, 0.08] ) = [0.01, 0.02] ∪ [0.251, 0.252] ∪ [0.501, 0.502] ∪ [0.751, 0.752] is no different from the measure of [0.04, 0.08]. That hypothesis suffices for the proofs of ergodic theorems. This transformation is measure-preserving.