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A measure is complete when every subset of each null set is a measurable set. Such a subset also has measure zero by monotonicity. The completion of a measure extends it to a complete measure by adjoining all subsets of measurable null sets to its sigma-algebra.

Lebesgue measure on the real numbers extends interval length: $\lambda ([a,b])=b-a$ for $a\le b$. Its restriction to the Borel sigma-algebra is the unique translation-invariant sigma-finite measure on that sigma-algebra normalized by $\lambda ((0,1])=1$. The full Lebesgue measure is the completion of a measure of this restriction; its domain is the Lebesgue measurable sets. Translation invariance follows by comparing a translated Borel measure on half-open intervals and applying the uniqueness theorem for measures. For the construction and interval properties, see MIT lecture 8.

Every singleton has Lebesgue measure zero: $\{x\}\subseteq [x-\epsilon ,x+\epsilon ]$ gives $0\le \lambda (\{x\})\le 2\epsilon$ for every $\epsilon >0$. Enumerating the countable set $\mathbb{Q}\cap [0,1]$ and applying countable additivity yieldsThus a dense set can be a null set, and a union of null sets indexed by an uncountable set can have positive Lebesgue measure. The distinction explains both the Dirichlet function example and why almost everywhere is weaker than everywhere.

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.