Limits of Events
"The estimate eventually holds." "The alarm rings infinitely often." These are sentences about infinitely many events at once — this subchapter builds the grammar to say them, and the theorems that let probability hear them.
In this subchapter
2.2.1Why single events are not enough
In 2.1 we proved the continuity properties of P: if events climb A₁ ⊆ A₂ ⊆ … then P(⋃n An) = limn P(An), and if they descend with P(A₁) < ∞ the mirror statement holds. Useful — but so far these are statements about well-behaved sequences. Real questions are messier. "Does the estimator eventually stay within ε of the truth?" is not an increasing sequence of events; it flickers. "Does the portfolio breach its risk limit infinitely often?" is a claim about the whole tail of a sequence, not about any single An.
The tool is two limit events — liminf and limsup of a sequence of sets — and the payoff is that every tail-question about a sequence of events is one of four events in disguise. This subchapter builds the grammar, proves the inequalities that connect the limits of events to the limits of their probabilities, and cashes both out immediately: the first Borel–Cantelli lemma, and the fact that "Xn converges" is a legitimate event.
2.2.2liminf and limsup: the two limit events
For a sequence of events A₁, A₂, … ⊆ Ω:
Read them aloud. liminf: "there is a start time m such that An happens for every n from m on" — eventually always. limsup: "for every start time m there is a later n with An happening" — infinitely often. The two events belong to the σ-algebra automatically: each is a countable sandwich of unions and intersections, exactly the closure clauses of 1.1.
A risk system fires an alert on day n if a loss Ln exceeds a threshold: An = {Ln > c}. Then limsup An is "there will be infinitely many breach days", and its complement is "only finitely many breaches" — equivalently ⋃m ⋂n≥m Anc, "from some day on, clean every day". Note the complement relation (limsup A)c = liminf Ac — De Morgan through the countable quantifiers — which is how the two events are two views of one fact.
The event limsup An can be huge (probability 1) even when every individual P(An) is tiny. Conversely it can be empty even when P(An) is close to 1 for every n. The limit events are about the pattern in time, not the size of any single slice. Section 5 makes the first phenomenon precise.
2.2.3Continuity of probability, the honest statements
From 2.1: if An ↑ A (increasing, union A) then P(A) = lim P(An); if An ↓ A (decreasing, intersection A) then P(A) = lim P(An), provided P(A₁) < ∞. Two remarks that make these theorems usable rather than decorative.
First, the finiteness condition in the descending case is load-bearing. We saw the tails marching away in 2.1: An = [n, ∞) under length, each of infinite measure, intersection empty. For probabilities the condition is automatic — P(A₁) ≤ 1 < ∞ — and this is one of the places where being a probability (not just a measure) quietly pays.
Second, the theorems are exactly what lets us compute limit events: to weigh limsup An, weigh the descending sequence Bm = ⋃n≥m An and take limits; each Bm is a countable union, weighable by additivity. The bridge between "pattern in time" and "numbers" runs through continuity, and we now cross it for good.
2.2.4Fatou for events
For any events A₁, A₂, …:
The probability of the limit event never overshoots the limit of the probabilities — in either direction.
Set Bm = ⋂n≥m An. Then Bm ↑ liminf An, and Bm ⊆ Am so P(Bm) ≤ P(Am). Continuity from below: P(liminf An) = limm P(Bm) = liminfm P(Bm) ≤ liminfm P(Am) — the middle step because P(Bm) is increasing, so its limit is its liminf. The right half is the same argument upside down: Cm = ⋃n≥m An ↓ limsup An, Am ⊆ Cm, continuity from above (legal: P(C₁) ≤ 1). The middle inequality liminf ≤ limsup is true for any real sequence. ∎
On [0,1] with length-probability, let An be the "moving spotlight": An = [0, 1/2] for odd n, [1/2, 1] for even n. Every P(An) = 1/2, so both liminf and limsup of the numbers are 1/2. But liminf An = ∅ (no point is in both halves forever) and limsup An = [0,1] (every point is caught infinitely often). Both inequalities strict, simultaneously: the probability of the limit event and the limit of the probabilities are different animals, and only under extra hypotheses (monotonicity, or independence — 6.3) do they meet.
2.2.5A first harvest: the first Borel–Cantelli lemma
The single most-used corollary of everything in this chapter — the tool that tames "infinitely often".
If ∑n P(An) < ∞, then P(limsup An) = 0: with probability one, An happens only finitely often.
For every m, Fatou's right half plus the union bound of 2.1 give
The left side does not depend on m; the right side is a tail of a convergent series, which marches to 0 as m → ∞. A number squeezed between 0 and something tending to 0 is 0. ∎
Daily breach probability P(An) = 1/n². The series ∑1/n² converges (to π²/6 — we only need finiteness), so infinitely many breach days has probability zero. Not because any single day is safe — day 1 breaches often — but because time exhausts the danger: the breaches must thin out. Contrast a constant P(An) = ε: the series diverges, the lemma is silent, and whether breaches persist is a question about dependence — answered by the second lemma in 6.3, under independence.
Take An = A, the same event repeated, with P(A) = ε > 0. Then ∑P(An) = ∞, yet nothing about "infinitely often" is forced by the divergence — here limsup An = A, probability ε, which can be anything in (0, 1]. The first lemma is a one-direction tool; the converse that fills the other direction needs independence, and that hypothesis is genuinely necessary — 6.3 does it honestly.
2.2.6Convergence is an event
Close with the promise made in 0.6: that statements about limits of random quantities are events — sets a probability can weigh. Here is the construction, once, in general.
Let X₁, X₂, … and X be real-valued random variables on (Ω, F, P) — meaning, from Chapter 4's perspective, that sets like {|Xn − X| > ε} are events. Then the set of ω where Xn(ω) → X(ω) is an event, and it equals
"Xn(ω) → X(ω)" says: for every tolerance 1/k, eventually all the Xn sit within 1/k of X(ω). "Eventually all" is exactly liminf: for each k, the inner ⋃m ⋂n≥m builds liminf { |Xn − X| ≤ 1/k }, an event by the closure clauses; the intersection over the countably many k stays in F. Running the ε–quantifiers over 1/k instead of all real ε > 0 is what keeps the intersection countable — the same density-of-rationals move as in 1.3. ∎
The Law of Large Numbers of 7.2 is the statement P({X̄n → μ}) = 1. For that sentence to mean anything, the set inside must be an event — which is exactly what we just built. In 0.6 we promised that frequencies need a foundation: this is the foundation's foundation. Every limit theorem in this course is a probability statement about a countable sandwich, and the grammar of this subchapter is what makes those sandwiches edible.
"Eventually" and "infinitely often" are events. Once that is true, every limit statement in probability is just arithmetic with countable sandwiches.
The axioms are alive: they compute limit events, and Borel–Cantelli is already earning its keep. Next we ask what a probability measure is made of: atoms — the indivisible positive-mass grains of a measure — and the theorem that a measure has at most countably many of them. Discrete probability done properly, and the first sharp line between the discrete world and the continuous one.