Week 7 Coursework Test

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