Measure-preserving transformations and invariance

A measurable transformation T preserves a measure when the preimage of every measurable event has the same measure as the event itself. In probabilistic language, applying T to a state distributed according to that measure leaves its distribution unchanged. This is an equality of marginal laws, not a statement that the state remains fixed. Equivalently the pushforward measure under T equals the original measure. Testing only a few selected events may suggest invariance but is a proof only when those events generate the sigma-algebra and an appropriate uniqueness argument applies.

The preimage definition works whether or not T is one-to-one. On the unit circle, doubling modulo one preserves Lebesgue probability even though two distinct points usually have the same image. Its preimage of an interval consists of two half-length pieces, whose lengths add to the original length. By extending this calculation from interval generators to measurable sets one obtains full preservation. An invariant probability measures the distribution of states, rather than assigning a unique predecessor to every state.

An invertible measure-preserving system has an inverse map that is measurable, at least after restricting to appropriate full-measure sets when working modulo null sets. Such a map is a probability-space automorphism. Only then can negative iterates be used as transformations without choosing among possible predecessors. Forward recurrence and the pointwise ergodic theorem do not need this stronger assumption, whereas the unitary formulation of the Koopman operator and a two-sided Lyapunov splitting will explicitly use it.

For a preserving map, composing observables with T defines an isometry on each finite Lp space. The function's values are rearranged without changing their distribution, so its norm is preserved. In L2 this operator preserves inner products as well. It is unitary only if it is also onto; invertible bimeasurable preservation supplies that property. A noninvertible doubling map gives an isometry whose range contains only observables that cannot distinguish the two points merged by doubling, so surjectivity should not be assumed.

For a permutation of finitely many states, an invariant probability must assign equal mass to every state within a given cycle. The total masses of different cycles can be chosen independently and then normalized to sum to one. Uniform probability on the whole state space is one possibility, but need not be the only one. These separate cycle masses are invariant information: the orbit never moves from one cycle to another. Later ergodic decomposition replaces this elementary picture by a measurable family of invariant component measures.

Measure preservation says nothing yet about whether a typical trajectory explores all statistically relevant states. A permutation with several cycles preserves its invariant probability but cannot move between the cycles. Even a single finite cycle has periodic correlations rather than the decay required for mixing. On an infinite counting-measure space, translation preserves measure while individual states can drift away forever. Thus recurrence, ergodicity, and mixing are additional conclusions with their own hypotheses, not alternate names for the defining preservation identity.