I dear friends. I have a question -
We take a count for the rationals in $[0,1]: Q \cap [0,1] = \uplus_{n=1}^{\infty} \{r_n\}$.
1) Does the sum: $\sum_{n=1}^{\infty}\frac{1}{2^k \cdot |x-r_k|^\frac{1}{2}}$ coverges almost everywhere in $[0,1]$ with respect to the one-dimensional lebesgue measure $m$?
2) If so, is the sum, call it $S$, is such that $S \in L^1([0,1])$?
3) Show (if $S$ exist almost everywhere) that $S \notin L^2(I)$, for every interval $I \subset [0,1]$ with $m(I) > 0$
Thank you so so much! P.S
Hints are also welcome :)