# 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

[itex](X, \mathcal{B}, T, m)[itex]

with the following structure:

[itex]X[itex] is a set,
[itex]\mathcal{B}[itex] is a [itex]\sigma[itex]-algebra over [itex]X[itex],
[itex]m:\mathcal{B}\rightarrow[0,1][itex] is a probability measure, so that [itex]m(X)=1[itex], and
[itex]T:X\rightarrow X[itex] is a measurable transformation which preserves the measure [itex]m[itex], i. e. each measurable [itex]A\subseteq X[itex] satisfies
[itex]m(T^{-1}A)=m(A).[itex]

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

[itex]m(T(A))=m(A)[itex]

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.

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