Let $X=\{0,1 \}^{\mathbb{Z}}$. So an element of $X$ is given by a sequence $(x_i)_{i\in \mathbb{Z}}$, where $x_i \in \{0,1 \}$.
Let $\mu_i$ be a probability measure on $(\{0,1 \}, \mathcal{P}(\{0,1 \}))$, where $\mu_i(\{0\})= \mu_i (\{1 \})= \frac{1}{2}$.
Define $\mu = \Pi_{i\in \mathbb{Z}} \mu_i$, and $B = \bigotimes_{i\in \mathbb{Z}} \mathcal{P}(\{0,1 \})$.
Let $T: X \rightarrow X$ be the Bernoulli shift defined by $T(x_i)=x_{i-1}$ for $i\in \mathbb{Z}$.
Show $T$ and $T^{-1}$ are measurable and $T$ is measure-preserving.
Attempt:
Let $A \in B$. To show that $T$ is measurable, I want to show that $T^{-1}(A)= \{x_{i+1} : (x_i)_{i\in \mathbb{Z}} \in A \} \in B$.
Define $H= \{A \subset X : T^{-1}(A) \in B \}$. I showed that $H$ is a $\sigma$-algebra. So $B \subset H$. Thus, $T$ is measurable.
To show that $T^{-1} $ is measurable, can I do the same thing using $\sigma$-algebras? I'm not sure about this part.
To show that $T$ is measure-preserving, I need to show that $\mu(A)= \mu(T^{-1}(A))$ for all $A\in B$. I don't have any clues for this part either.
Any help will be appreciated.
Thank you!