I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma algebra of a certain metric.
It's the following: In E, we define the distance of two sequences $ x, y $ as the infinite sum of $\sum{|x_n-y_n|2^{-n}}$ which easily makes E a compact metric space. A subset S of E is a tail if for every $y$ for which there is a $x$ in S such that $x, y$ are eventually the same, then $y$ is in S. Then we must show that every tail is borel. But it does not seem true. I can reduce tails to arbitrary unions of closed sets, but not for countable unions. What am I missing?
A non principal ultrafilter on the set of natural numbers is a tail set but it is far from Borel. What you are missing is that you need to make use of the fact that your set is Borel (or Lebesgue measurable) so that you can apply something like Lebesgue density theorem.