I have a question concerning this Wikipedia on metric entropy.
There, the following object is defined: $$ \bigvee_{n=0}^NT^{-n}Q=\{Q_{i_0}\cap T^{-1}Q_{i_1}\cap\ldots\cap T^{-N}Q_{i_n}: \textrm{where }i_l=1,\ldots,k, l=0,\ldots,N\} $$ and each element has positive measure.
I am wondering if this is also defined if $N\to\infty$ and how it looks like: Does all elements still have positive measure for $N\to\infty$? In other words, what is $$ \bigvee_{n=0}^{\infty}T^{-n}Q? $$
EDIT: I think $\bigvee_{n=0}^{\infty}T^{-n}Q$ is usually defined to be the smallest $\sigma$-algebra which contains $\bigvee_{n=0}^N T^{-n}Q$ for each $N\in\mathbb{N}$ (see, for example, here, Definition 5).
Indeed, $\bigvee_{n=0}^{\infty}T^{-n}Q$ denotes the $\sigma$-algebra generated by the elements of $\bigvee_{n=0}^NT^{-n}Q$ for all $N\in\mathbb N$. It may happen for example that all elements of $\bigvee_{n=0}^{\infty}T^{-n}Q$ besides the empty set have positive measure or that only the empty set has zero measure. That depends on each measure and on each map.
For example, take $T(x)=2x\bmod1$ on $[0,1]$ with the Lebesgue measure. The Borel $\sigma$-algebra has the cardinality of $\mathbb R$ and so for example for $Q$ composed of $[0,\frac12]$ and $[\frac12,1]$ it is impossible that the sets in $\bigvee_{n=0}^NT^{-n}Q$ for all $N\in\mathbb N$ contain all the elements of the $\sigma$-algebra.
Two corrections on your post: you need to erase $l=0,\ldots,N$ in the definition of $\bigvee_{n=0}^NT^{-n}Q$ and it is false in general that all the sets in $\bigvee_{n=0}^NT^{-n}Q$ have positive measure (note that in fact some may even be empty).