Quasi-ergodicity

Remarks

Here $\triangle$ is the symmetric difference, so the condition $\mu(T^{-1}(A) \mathbin{\triangle} A) = 0$ says that $A$ is invariant up to a null set. If $T$ preserves $\mu$ and $\mu$ is a probability measure, quasi-ergodicity is the same as being an ergodic transformation. Example: let $\mu$ be a probability measure on $[0,1)$ with a strictly positive density $\rho$ with respect to Lebesgue measure $\lambda$, and let $T(x) = x + \alpha \bmod 1$ with $\alpha$ irrational. The measures $\mu$ and $\lambda$ have the same null sets, and $T$ preserves $\lambda$, so $T$ is quasi-measure-preserving for $\mu$. An almost invariant set for $\mu$ is almost invariant for $\lambda$, so it has $\lambda$-measure $0$ or $1$ by ergodicity of the rotation, hence $\mu$-measure $0$ or $1$. Non-example: the identity map of $[0,1]$ with Lebesgue measure is quasi-measure-preserving, but $A = [0, 1/2]$ is invariant with $0 < \lambda(A) < 1$. In older physics texts the "quasi-ergodic hypothesis" means something else: that a trajectory comes arbitrarily close to every point of its energy surface.