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(i) SUM_n (P(A_n)) < oo => P( limsup_n->oo (A_n) ) = 0

(ii) If in addition the A_n are independent then

P( limsup_n->oo (A_n) ) <1 => SUM_n (P(A_n)) < oo

Does that mean if the A_n are independent then P( limsup_n->oo (A_n)) must be either 0 or 1??

If so, why bother using "<1" in (ii) and not just use "=0" instead?

If not, then when it is strictly between 0 and 1 we have

from (ii) that SUM_n (P(A_n)) < oo

and then from (i) we get P( limsup_n->oo (A_n) ) = 0, a contradiction.