An approach to Borel-Cantelli for the $l^p$ Let $\mu$ be a non-negative measure and $\{E_{k}\}$ a sequence such that $\sum \mu(E_k)^p<\infty $ then show that $F=\lim \ \sup E_{k}=\cap_{k=1}^{\infty}\cup_{n\geq k}E_n$ has $\mu$ measure zero.
 A: When $p\in (0, 1]$, this obviously reduces to the classical Borel-Cantelli lemma. When $p > 1$, the result is not true in general. For a counterexample, consider the following construction (we take $\mu$ to be the Lebesgue measure on $\mathbb{R}$).
So basically the idea is to cover $(0,1)$ by a sequence $\{F_j\}$, such that $\mu(F_j) = 1/j$, while for arbitrarily large $k$, $\{F_j\}_{j\geq k}$ still cover $(0,1)$. 
For any $n\geq 1$, let $\mathcal{C}_n = \{E_{i,n}\}_{i\geq 1}$ be a finite open cover of $(0,1)$ such that $\mu(E_{i,n}) = 1/{(n-1+i)}$ (which can of course be constructed since $\sum_{k=1}^\infty 1/k$ diverges). Now construct a countable sequence of sets recursively as follows. First take $G_1 = C_1$. For $j > 1$, if $m$ is such that $\min_{E\in C_{j-1}}\mu(E) = 1/m$, set $G_j = C_{m-1}$. Let $F = \bigcup_{j}G_j$. Now we have:
$$
\sum_{E\in F}\mu(E)^2 < \infty.
$$
Enumerate elements in $F$ as $F_1, F_2, \dots$ in order of decreasing measure. Then we get
$$
(0,1)\subset \bigcap_{j\geq k}F_j,
$$
for every $k\geq 1$. 
