The concept of Cylinder Set

Question

Stein Real Analysis chapter 6 exercise no.15 Introduces Cylinder sets as $$\{x \in (x_j), x_j \in E_j , E_j \in M_j, \mbox{but }E_j =X_j \mbox{ for all but finitely many } j\},$$ First of all I am confused with the part $E_j=X_j$ for all but finitely many $j$. I guess it means finite number of $E_j$ may be equal to $X_j$ not all of them or countable number of $E_j$ ;However, it seems pointless to not let all $E_j$ equal to $X_j$. I also can not imagine the shape of this set; So we have so many elements of infinite number of measurable sets what it gives to us, How it is important?

Answer 1

It means that for all but finitely many $j$s we have $E_j=X_j$. So the cylinder sets are finite intersections of sets of the form $E_1\times X_2\times X_3\times\cdots$ or $$X_1\times\cdots\times X_{j-1}\times E_j\times X_{j+1}\times X_{j+2}\times\cdots$$ for some $j>1$. Notice that the topology, the $\sigma$-algebra, etc is generated by the sets of the form $$E_1\times\cdots\times E_j\times X_{j+1}\times X_{j+2}\times\cdots$$ with $j\ge1$.

PS: Since you use the tag "ergodic theory", let me add that in that case you should take the discrete topology on each factor.