I am currently attempting to work through Walters' text on Ergodic theory and in the proof of the list of equivalent conditions for ergodicity they state
$T^{-n}B\triangle B\subset\cup_{i=0}^{n-1}T^{-(i+1)}B\triangle T^{-i}B$ after only having assumed that $\mu(B\triangle T^{-1}B)=0$. The measure of this set is zero, but there could still be points in it. Isn't it true that any point, $x$, in this null set would not be in the above union if, e.g. $Tx$ had a return time larger than n?
Are we meant to have taken the original definition of ergodicity ($T^{-1}B=B\implies\mu(B)\in\{0,1\}$) to be up to null sets in its hypothesis? I suppose I understand not wanting to write "up to a set of measure 0" over and over again throughout the text, but using the subset notation without making this allowance seems slightly egregious so I feel as though I'm missing something.
The inclusion holds at every point and no null sets are involved in it. Let $x$ belong to the left side. Then there are two possibilites: $x \in B, T^{n}x \notin B$ or $x \notin B, T^{n}x \in B$. Suppose $x \in B, T^{n}x \notin B$. Consider $T^{k}x$ for $k=1,2,\cdots,n$. There is a largest $k$ such that $T^{k}x \in B$. Note that $k <n$. It follows that $T^{k}x \in B$ but $T^{k+1}x \notin B$. Hence $x \in T^{-k}B \Delta T^{-k-1}B$. we have proved that $x$ belongs to the right side. A similar argument holds for the case $x \notin B, T^{n}x \in B$.