Say A is the set of all Real numbers on $[0,1]$ whose decimal expansion contains an infinite number of 8s. I am trying to prove the measurability of this set.
I realize that this is the set of numbers for which 8 is in the $\limsup$ of their decimal expansion. I also know that $\limsup$ of any set is measurable, as it is just an infinite union of infinite intersections. I am stuck at how to use this fact to show that this set is measurable.
Another approach would be to construct nested intervals around each each real number that contains an infinite number of 8s and then take the intersection. The union of all these intersections would be my set--but I don't think that is countable, so I can't use it to prove measurability.
Any hints would be more than welcome. Thank you.
$A^c$ is the set of real numbers $x$ so that the decimal expransion of $x$ contains only finitely many $8$'s. Let
$$ B_k = \{x\in [0,1]:\ \text{the k-th entry of the decimal expansion of x is not 8}\}$$
Then $B_k$ is measurable, being a union of intervals. Then we have
$$A^c = \bigcup _{k=1}^\infty \bigcap_{m=k}^\infty B_m$$
and so $A^c$ is measurable.
(An interesting discussion here)