The following is from Walters' Ergodic Theory book :
Blue : 1- How $m$ is defined based on $\mu$? 2- If spaces are all sets of integers where does measurable rectangles come from?! 3- How $m(T^{-1}A) = m(A)$ holds?
Red : 4- How T is measure preserving? 5- How $T^{-1}$ is measure preserving?
First $X$ is just the set of (bi infinity) sequences of element of $Y$ and $T$ is just the shift in the sequence, you change the indices by $n \mapsto n+1$.
Now the measure $m$ is as follow $m(.....i....)=p_i$ that is the measure of the set of sequences having $I$ at a given position is $p_i$. And then $$ m(...i_1 i_2 - i_l....)=p_1p_2-p_l $$ Where "$...$" mean any number and "$-$" mean given number.
This answer 1 and 2.
3, 4 and 5 are now just verifications.
I hope it is clear, comment if you need more details.