Problem with understanding the analytic continuation of zeta function.
What is the analytic continuation of the Riemann Zeta Function
$P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$ Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) &gt; 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second Borel-Cantelli out because we don't know if the $A_n$'s are independent?)  Suppose $\forall A \in \mathfrak{F}$ s.t. $P(A) &gt; 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$, but $P(\limsup A_n)&lt;1$. My prof gave us this convenient hint: Show $\exists M &gt; 0$ s.t. $P(\bigcap_{j=M}^{\infty} A_j^{c}) &gt; 0$.  $1 &gt; P(\limsup A_n)$  $1 - P(\limsup A_n) &gt; 0$  $P(\liminf A_n^{c}) &gt; 0$  By definition of $\liminf A_n^{c}$, $\exists M &gt; 0 $ s.t. $P(\bigcap_{j=M}^{\infty} A_j^{c}) &gt; 0$. (Side questions 2 and 3 Isn't this =1? ...Wait, is that what we're trying to prove? 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