Discover published sets by community

A transformation $T: X \to X$ on a measure space $(X, \mathcal{B}, \mu)$ is said to be measure-preserving if for every set $A$ in the sigma-algebra $\mathcal{B}$, $\mu(T^{-1}(A)) = \mu(A)$. Such transformations keep the measure of sets invariant under the application of $T$.

The Birkhoff Ergodic Theorem forms the basis for connecting time averages and space averages which allows one to infer long-term statistical properties of dynamical systems from spatial distributions. This has practical implications in fields like statistical physics, where time invariance of ensemble averages is a foundational assumption.

A sequence of functions $f_n$ converges almost everywhere to a function $f$ if $f_n(x) \to f(x)$ for all $x$ in the space, except possibly for a set of measure zero. In the context of the Birkhoff Ergodic Theorem, it means the time averages converge to the space average for all points except a set of measure zero.

A measure-preserving transformation $T$ is ergodic if every $T$-invariant set (a set $A$ such that $T^{-1}(A) = A$) has measure zero or one. Ergodicity implies that the system cannot be decomposed into simpler, invariant subsystems and has implications for the uniqueness of space averages.

Invariant sets are a key concept in determining the ergodicity of a transformation. If all invariant sets for a transformation have measure zero or one, the transformation is ergodic. This is important as it indicates the transformation cannot be decomposed into invariant subsystems with intermediate measures.

The Birkhoff Ergodic Theorem states that for a measure-preserving transformation on a probability space, time averages of an integrable function converge almost everywhere to space averages. This implies that for almost all points, the long-run average of a function along the trajectory of the point equals the average of the function across the space.

The time average of a function $f$ with respect to a transformation $T$ and a starting point $x$ is given by $\lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} f(T^i(x))$. It represents the average value of the function along the orbit of $x$ under repeated applications of $T$.