Event

An event is a measurable set in a probability space: a question with a yes-or-no answer for each outcome. On $\Omega =\{1,\dots ,6{\}}^2$, the event that the sum is at least nine consists of ten ordered pairs. Its probability is therefore $10/36=5/18$ under independent uniform dice. Saying that an event occurred means the actual outcome belongs to that set.

For events $A_n$ in a probability space, the infinitely-often event isIts complement of an event is ${\bigcup }_{m\ge 1}{\bigcap }_{n\ge m}{A}_n^c$: after some outcome-dependent index, no more events occur.^[ref]

Deleting finitely many events leaves the limsup of events unchanged. It is therefore a tail event of the sequence of indicator random variables $1_{A_n}$, 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 $1_{A_n}\to 0$ precisely when only finitely many events occur, connecting the First Borel-Cantelli lemma to almost sure convergence and the existing complete convergence criterion.

Let $(a_n)$ be a nondecreasing real sequence tending to $+\infty$, and let $(x_n)$ be any real sequence. The running maximum exceeds $a_n$ infinitely often exactly when $x_n>a_n$ infinitely often. Applying this pathwise to random variables gives the displayed event equality, without any independence assumption.

One implication follows from $x_n\le {\max }_{k\le n}{x}_k$. For the other, if $x_n\le a_n$ from some $N$ onwards, every later value with $N\le k\le n$ satisfies $x_k\le a_k\le a_n$. The finite prefix also lies below $a_n$ for all sufficiently large $n$, because $a_n\to \infty$. Therefore the entire running maximum eventually stays below the threshold.

Divergence of the thresholds prevents one large early observation from creating infinitely many crossings. Monotonicity ensures that a later running-maximum crossing is controlled by the corresponding earlier individual threshold. These two conditions explain why the resulting event belongs to the tail sigma-algebra when the thresholds are deterministic.

For any sequence of events, its limsup of events satisfiesSet $C_N={\bigcup }_{n\ge N}{A}_n$. These are decreasing events with intersection ${\mathrm{limsup}}_n{A}_n$. Continuity of probability from above givesThis is also the bounded-indicator form of the reversed Fatou lemma. The inequality can be strict: independent fair-coin head events each have probability $1/2$, while their limsup of events has probability $1$ by the Second Borel-Cantelli lemma.

The event that $A$ does not occur has probability $1-P(A)$. Conditioning on an event positively associated with $A$ negatively associates its complement with $A$.

Two events are disjoint when they cannot both occur. For a finite family of pairwise disjoint events, the probability of their union is the sum of their probabilities, so that sum is at most one. This condition differs from the condition defining independent events.