Invariant sigma-algebras and ergodic decomposition

Let X be a standard Borel space, let T be a Borel measurable map, and let mu be a T-invariant probability. Standard Borel means the measurable structure is isomorphic to that of a Borel subset of a Polish space; finite and countable discrete spaces are included. One may complete the probability space when discussing almost-sure identities. These regularity assumptions permit regular conditional probabilities. An arbitrary measurable space need not admit the probability kernels used in this statement. No invertibility assumption is needed here.

The invariant sigma-algebra consists of events unchanged by preimage modulo null sets. A conditional probability kernel assigns to almost every point x a probability mu-x whose integral against a fixed integrable observable gives its conditional expectation on this sigma-algebra. For each Borel event, the assigned probability is a measurable function of the component information. Equalities are almost-sure identities; they do not specify a canonical measure for every exceptional point. Versions of the whole kernel are unique modulo null sets when interpreted through a countable determining class.

The ergodic decomposition theorem says these conditional probabilities can be chosen invariant and ergodic for almost every x. Their dependence is invariant along typical trajectories, so mu at Tx equals mu at x almost everywhere. Invariance of each component says it is a legitimate dynamical probability law; ergodicity says it has no further nontrivial invariant event under that law. This is stronger than merely decomposing the original measure into arbitrary probabilities, which is possible in many irrelevant ways and need not respect the dynamics.

The theorem is invoked here, not proved by identifying orbit sets. A component need not be one orbit, and quotienting by individual orbits need not produce a well-behaved measurable parameter space. Instead the countable structure of a standard Borel space lets conditional probabilities and a simultaneous determining family interact with the pointwise theorem. Birkhoff then identifies time averages with component integrals. This explains both why the component law is dynamically meaningful and why measurability is a substantive hypothesis rather than a decorative assumption.

For a finite permutation, each positive-mass cycle carries its unique uniform invariant probability. The original invariant law assigns a total weight to each cycle and is the corresponding mixture of those uniform laws. A function's component integral is simply its cycle average. Averaging these means using the cycle weights recovers the spatial mean. Forgetting the weights, or dividing by the total number of points when cycles have different masses, generally produces a different and incorrect expectation.

For the identity transformation on the unit interval with Lebesgue probability, every event is invariant. The component probability at x is the point mass at x, although every singleton has Lebesgue mass zero. The mixture of these point masses is Lebesgue measure, and each trajectory remains at its starting point. Conditioning by a formula that divides by the singleton's Lebesgue mass is impossible. A probability kernel nevertheless gives an exact decomposition, showing why continuously indexed components cannot be handled solely with positive-probability event conditioning.