\[ \newcommand{\C}{\mathbb{C}} \newcommand{\cont}{\operatorname{\mathsf{C}}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\sph}{\mathsf{S}^\mathsf{1}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\orb}{\mathop{\mathsf{orb}}} \newcommand{\symdiff}{\mathop\triangle} \newcommand{\lp}[1][2]{\operatorname{\mathscr{L}^{\mathsf{#1}}}} \newcommand{\Lp}[1][2]{\operatorname{\mathsf{L}^{\!\mathsf{#1}}}} \]

1. Give an example of a point $x \in \{0,1\}^\N$ for which \[ \lim_{r \to \infty} \dfrac{1}{2^{2r}} \sum_{n=1}^{2^{2r}} x(n) \qquad \lim_{r \to \infty} \dfrac{1}{2^{2r+1}} \sum_{n=1}^{2^{2r+1}} x(n) \] both exist and are distinct.
2. The strong law of large numbers states for the $(p,1-p)$ coin measure $\mu_p$ that the set \[ F_p = \left\{ x \in \{0,1\}^\N : \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n) = p \right\} \] has measure 1.
   1. Prove that $p \ne q$ implies $F_p \cap F_q = \emptyset$.
   2. We have $\mu_p(X) = 1$ for all $0 \le p \le 1$. Briefly explain why there is no contradiction in writing $X$ as an uncountable union of sets $F_p$ each having measure 1.