Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 26J (Probability and Measure)

A probability measure $\mu$ is $T$-invariant whenEquivalently, the pushforward measure satisfies $T_{\ast }\mu =\mu$, so $T$ is a measure-preserving transformation.

Assuming $\mu$ is $T$-invariant, $T$ is an ergodic transformation when every measurable set invariant modulo null sets is trivial:Equivalently, the invariant sigma-algebra is trivial modulo null sets.

The Birkhoff ergodic theorem states that if $T$ is a measure-preserving transformation and $f\in L^1(\mu )$, thenwhere $\mathcal{I}$ is the invariant sigma-algebra. The limit has the same integral as $f$. If $T$ is an ergodic transformation, the limit is the constant ${\int }_{X}fd\mu$ almost everywhere.

Take the unit circle $X=\mathbb{R}/\mathbb{Z}$ with its Borel sigma-algebra and normalized Lebesgue measure, and letThis is the doubling map. Its two inverse branches halve length, so Lebesgue invariance of the doubling map gives $\mu (T^{-1}A)=\mu (A)$. Moreoverfor every point of the circle.

For every $x\in X$, invariance givesThus $T^{-1}\{x\}$ is nonempty, so $T$ is a surjective function. A surjection from a finite set to itself is a bijection. ThereforeThis is the finite full-support measure-preserving map is bijective principle.

Because $X=\mathbb{N}$ is a countable set and $\mu (X)=1$, some $x\in X$ satisfies $\mu (\{x\})>0$. Invariance and the inclusion $\{T^{n}x\}\subseteq T^{-1}\{T^{n+1}x\}$ giveIf the forward orbit contained infinitely many distinct points, their disjoint singletons would each have measure at least $\mu (\{x\})>0$, contradicting that $\mu$ is a probability measure. The orbit therefore eventually enters a finite periodic orbit $Y$.

Now $T(Y)=Y$, so $Y\subseteq T^{-1}Y$. Since $\mu$ is invariant,Thus $Y$ belongs to the invariant sigma-algebra modulo null sets. It has positive measure, so the fact that $T$ is an ergodic transformation forcesThis is the ergodic invariant probability on a countable state space has finite cyclic support.