Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 26K / c / ii

The event $\{X_n=1\}$ is $T^{-n}(0,1/2)$. Since $T^n$ has $2^n$ affine branches of slope magnitude $2^n$, this preimage is, up to endpoints, a disjoint union of $2^n$ intervals of length $2^{-(n+1)}$ and has measure $1/2$.

More generally, for every binary word $(a_0,\dots ,a_{N-1})$, its itinerary cylinderis, up to finitely many endpoints, one monotonicity interval of $T^N$ and has measure $2^{-N}$. Thereforeso the $X_n$ are independent and identically distributed random variables, each with the Bernoulli distribution of parameter $1/2$.

The finite itinerary cylinders have diameters at most $2^{-N}$. Their endpoints form a dense set, so their unions generate the Borel sigma-algebra after completion by null sets. Thusin the standard probability-space convention that σ-algebras are identified modulo null sets.

There is a small literal endpoint defect in the uncompleted formulation of the question: $0$, $1$, and $2/3$ all have the all-zero itinerary, so $\sigma (X_0,X_1,\dots )$ cannot separate these Borel singletons. Consequently the displayed equality is false as an equality of raw Borel σ-algebras with the stated open interval, but it is true after completion and modulo null sets, which is the version used in the ergodicity argument.