Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 22J (Probability and Measure)

A measure-preserving transformation is a measurable map $T:E\to E$ satisfying $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. An invariant set satisfies $T^{-1}A=A$ (or equality modulo null sets, in the measure-theoretic convention).

Preimages preserve complements and countable unions. Therefore the whole space is invariant, the complement of an invariant set is invariant, and a countable union of invariant sets is invariant. Thus the invariant sets form a $\sigma$-algebra. The same proof works modulo null sets because a countable union of null exceptional sets is null.

For a Borel set $A\subset [0,1)$, the two inverse branches of the doubling map give the disjoint setsEach has half the Lebesgue measure of $A$, by translation and scaling invariance. Hence $\boxed{\mu (T^{-1}A)=\mu (A)}$. The piecewise affine map is Borel measurable, so it is a measure-preserving transformation. The printed one-dimensional “baker's map” is this doubling map.

For a probability distribution $m$ on $\mathbb{R}$, take the canonical sample space $\Omega ={\mathbb{R}}^{\mathbb{N}}$, the product of the Borel $\sigma$-algebras, and the product measure $\mathbb{P}=m^{\otimes \mathbb{N}}$. The coordinate functions $X_j(\omega )={\omega }_j$ have joint cylinder probabilitiesThe Kolmogorov extension theorem supplies this measure from the consistent finite-dimensional distributions, and uniqueness follows from the generating cylinder sets. Thus the coordinates are independent and identically distributed; the same construction works for a given standard Borel state space.

The one-sided Bernoulli shift is $T({\omega }_1,{\omega }_2,\dots )=({\omega }_2,{\omega }_3,\dots )$. It is measurable and preserves all cylinder probabilities. The uniqueness theorem for probability measures extends this to the product $\sigma$-algebra, proving measure preservation.

To prove ergodicity, let $A$ be invariant modulo null sets and approximate it in probability by an event $C$ depending on only the first $r$ coordinates, with $\mathbb{P}(A△C)<ϵ$. Such approximations follow from the fact that finite-coordinate cylinder events generate the product $\sigma$-algebra. For $n\ge r$, independence gives $\mathbb{P}(C\cap T^{-n}C)=\mathbb{P}(C{)}^2$. Invariance and measure preservation giveAlso $|\mathbb{P}(A{)}^2-\mathbb{P}(C{)}^2|\le 2ϵ$. Letting $ϵ\downarrow 0$ proves $\mathbb{P}(A)=\mathbb{P}(A{)}^2$, so $\boxed{\mathbb{P}(A)\in \{0,1\}}$. This is exactly ergodicity.