Considering $X = \{0,1\}^\mathbb{N}$, the space of infinite sequences of $0$'s and $1$'s, and the $\sigma$-algebra generated by the cylinders sets $[x_1,x_2,…,x_n]$, where $x_i \in \{0,1\}$.
I want to show that the set of all $x\in X$ such that $\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1}{2}$ is in this $\sigma$-algebra.
I'm a little bit lost on this one, I'm not sure where to start.
I know the definition of a convergent sequence, and I know that I should be able to construct the set of such sequences from unions, intersections and complements of the cylinders.
Any idea would be of great help!
Best Answer
Let $\displaystyle E_{n,k}:=\left\{x\in X:\left|\frac1n\sum_{i=1}^nx_i-\frac12\right|\le\frac1k\right\}$. Clearly $E_{n,k}$ is a cylinder because \begin{align*} x\in E_{n,k}&\iff\frac{n(k-2)}{2k} \le\sum_{i=1}^nx_i\le\frac{n(k+2)}{2k}\\ &\iff(x_1,\ldots,x_n)\in\left\{b\in\{0,1\}^n:\left\lceil\tfrac{n(k-2)}{2k}\right\rceil\le b_1+\cdots+b_n\le\left\lfloor\tfrac{n(k+2)}{2k}\right\rfloor\right\}\!. &\end{align*} Now $$\left\{x\in X:\lim_{n\to\infty}\frac1n\sum_{i=1}^nx_i=\frac12\right\}=\bigcap_{k\ge1}\bigcup_{\vphantom km\ge1}\bigcap_{\vphantom kn\ge m}E_{n,k}.$$