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Definition
The expectation of an integrable observable under the chosen probability measure.
The space average of an integrable observable on is its expectation
It averages over states distributed according to . For an event indicator , this average is .
A time average samples one orbit. The two notions are defined independently; Birkhoff's theorem identifies typical limiting time averages with space averages when the system is ergodic.
An observable on a measurable dynamical system is a measurable function (or ). In probability-preserving dynamics it is usually considered modulo almost-everywhere equality in a specified function space, such as , , or .
For an observable and a probability-preserving transformation , its time average over steps is
Compare this orbit sample with the space average under .
For a probability-preserving transformation and , the Birkhoff pointwise ergodic theorem states
Here is the invariant sigma-algebra. Invertibility is not required. If is ergodic, the limit is .