= Limsup of events
{title2=$\limsup_{n\to\infty}A_n$}
= Infinitely often event
{synonym}
For <events> $A_n$ in a <probability space>, the infinitely-often <event> is
$$
\limsup_{n\to\infty}A_n
=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}A_n
=\{\omega:\text{for every }m\text{ there is }n\geq m\text{ with }\omega\in A_n\}.
$$
Its <complement of an event> is $\bigcup_{m\geq1}\bigcap_{n\geq m}A_n^c$: after some outcome-dependent index, no more <events> occur.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§2.3 opening definition and infinitely-often notation, printed pp. 67–68]{ref}
Deleting finitely many <events> leaves the <limsup of events> unchanged. It is therefore a <tail event> of the sequence of <indicator random variables> $\mathbf1_{A_n}$, whether or not those <random variables> are <independent>. For <independent events>, the <Kolmogorov zero-one law> restricts its <probability> to zero or one; the <Borel-Cantelli lemmas> determine which alternative holds from the <series> of <probabilities>. Also $\mathbf1_{A_n}\to0$ precisely when only finitely many <events> occur, connecting the <First Borel-Cantelli lemma> to <almost sure convergence> and the existing <complete convergence> criterion.
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