Let $\Sigma=\{1,2\}^{\mathbb{N}}$ and $A$ be the sigma-algebra generated by the cylinders sets $\{w∈Ω|∀s∈S,w_s=ϵ_s\}$ with $S⊂\mathbb{N}$ finite and $ϵ_s∈\{0,1\}$ . Let $p∈(0,1)$ . We take product measure with density $p$ on $(Ω,A)$ :$μ=∏_{n∈\mathbb{N}}μ_n$ where $μ_n$ is Bernoulli measure on $\{0,1\}$ given by $μ_n(w_n=1)=p$ , $μ_n(w_n=2)=1-p$.
Let $\mu_{1/2}$ be the corresponding product measure if we choose $p=1/2$.
Let $\mathbf{i}=(i_1,i_2,\dots)\in \Sigma$.
Let $\mathbf{i}|_n = (i_1,...,i_n)$, $n \in \mathbb {N}$.
For $ n \in \mathbb{N} $, define \begin{equation*} ||\mathbf{i}|_n||_j := \text{Number of occurrences of}~~ j ~~\text{in}~ (i_1,\dots,i_n),~j=1,2. \end{equation*}
Then $||\mathbf{i}|_n||_1+||\mathbf{i}|_n||_2=n.$
Set $x_n^{\mathbf{i}}=~||\mathbf{i}|_n||_1-||\mathbf{i}|_n||_2 =~2 ||\mathbf{i}|_n||_1-n$, for $n \in \mathbb{N}$.
If $A=\{\mathbf{i}\in \Sigma:~ \text{The sequence}~~(x_n^{\mathbf{i}})_{n\in \mathbb{N}}~~\text{is bounded above} \}$ $\subset \Sigma$.
Question: What is $\mu_{1/2}(A)$ ?
Conjecture: $\mu_{1/2}(A)=1$.
Any reference to Bernoulli product measure and Symbolic space is very much appreciated.
Being a Bernoulli product measure means that the Random variables $\{ i_n \}_{n\in \mathbb{N}}$ are i.i.d . That means that any string with positive probability of concurring will occur. Since $p$ and $1-p$ are both non-zero, any string has a positive probability of occurring.
Since you can write $A=\cup_{m\in \mathbb{N}} A_m$ where $A_m:=\{ \mathbf{i}\in \Sigma:~ x_n^{\mathbf{i}}\leq m \; \text{for all} \; n\in \mathbb{N} \}$, and $\mu(A_m^c)=1$, you get that $\mu(A)=0$.