$$$$

Put $X=\{0,1{\}}^{\mathbb{N}}$. Write ${\mu }_p$ for the $(p,1-p)$ coin measure.

1. What is ${\mu }_p([01])$?
2. What is ${\displaystyle \int 1_{[01]}\mathsf{d}{\mu }_p}$?
3. Define $Z_1(x)=x(1)x(2)$. What is ${\displaystyle \int Z_1\mathsf{d}{\mu }_p}$?
4. What is ${\mu }_p(T^{-1}([01]))$?
5. Define $Z_2=Z_1\circ T$. What is ${\displaystyle \int Z_2\mathsf{d}{\mu }_p}$?

For $n\ge 2$ define $Z_n=Z_1\circ T^{n-1}$. Define $$Y_n=Z_n-\int Z_n\mathsf{d}{\mu }_p$$ for all $n\in \mathbb{N}$.

6. Calculate ${\displaystyle \int Y_n\mathsf{d}{\mu }_p}$ for all $n\in \mathbb{N}$.
7. Calculate ${\displaystyle \int Y_n{Y}_{n+2}\mathsf{d}{\mu }_p}$ for all $n\in \mathbb{N}$.
8. Calculate ${\displaystyle \int Y_n{Y}_{n+1}\mathsf{d}{\mu }_p}$ for all $n\in \mathbb{N}$.
9. For which $a,b,c,d\in \mathbb{N}$ is ${\displaystyle \int Y_a{Y}_b{Y}_c{Y}_d\mathsf{d}{\mu }_p}$ equal to zero?
10. Prove the complement of $$\{x\in X:\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}x(n)x(n+1)\mathsf{\text{ exists}}\}$$ has zero measure with respect to ${\mu }_p$ and identify the limit.