MathDB

Consider the space

\{0,1 \} ^{N}

with the product topology (where

\{0,1 \}

is a discrete space). Let

T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}}

be the left-shift, ie

(Tx) (n) = x (n+1)

for every

n \in \mathbb {N}

. Can a finite number of Borel sets be given:

B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N}

such that

\left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \}

the

\sigma

-algebra generated by the set system coincides with the Borel set system?

Problem Statement