For T measure-preserving and f ∈ L¹(μ), the time average (1/n) Σ₀ⁿ⁻¹ f∘T^k converges a.e. to a T-invariant function ƒ*. If T is ergodic, ƒ* = ∫f dμ (time average = space average).
For T measure-preserving and f ∈ L¹(μ), the time average (1/n) Σ₀ⁿ⁻¹ f∘T^k converges a.e. to a T-invariant function ƒ*. If T is ergodic, ƒ* = ∫f dμ (time average = space average).