Events Before Probabilities: The σ-Algebra
A probability measure takes a set as input. Before we can talk numbers, we must decide which sets get to stand in line. That decision is the whole content of this subchapter.
In this subchapter
1.1.1Probabilities eat sets: why the domain matters
From Chapter 0 we know two things that the 101 course never told us. First, on a serious sample space, some subsets cannot be given a probability at all — the Vitali construction produces a delegate set V such that no length-like assignment on all subsets can exist. Second, the statements we care about — "the coin eventually shows heads", "the average converges" — are built from events using countably many unions, intersections and complements, so the collection of events must survive those operations.
So a probability model is a function P from some collection of subsets of Ω to numbers. The discipline question is: what properties must this collection have for the theory to work? Answering that is our job here, and it deserves care: get the domain wrong and everything downstream — conditioning, expectation, limit theorems — is built on sand.
The σ-algebra is not the place where probability gets restrictive. It is the place where probability gets honest about what it can measure.
1.1.2The definition, clause by clause
Let Ω be a set (our sample space). A collection F of subsets of Ω is a σ-algebra (sigma-algebra; the σ is a Greek lowercase "s", for somma — sums, countably many of them) when it satisfies three clauses:
- Containment: Ω ∈ F and ∅ ∈ F. Here ∅ is the empty set, the set with no elements; ∈ means "is a member of the collection".
- Complements: if A ∈ F, then its complement Ac = {ω ∈ Ω : ω ∉ A} is also in F. The complement is "everything that is not in A"; ω (lowercase omega) is the standard name for a single outcome.
- Countable unions: if A₁, A₂, A₃, … ∈ F (a sequence of them), then ⋃n An ∈ F, where ⋃n An is the set of outcomes lying in at least one of the An.
The pair (Ω, F) is called a measurable space, and the members of F are called the events, or the measurable sets.
Each clause is a demand written by a failure you have already met. Walk through them as experiments, not as bureaucracy.
"Something happens" is the event Ω; "nothing happens" is ∅. A model that cannot assign a probability to the statement the experiment produced an outcome (probability 1, naturally) and to none of the outcomes occurred (probability 0) is not a model of anything. In the dice space, "sum between 2 and 12" is all of Ω, probability 1 — and you use this triviality every time you check a probability table sums to 1.
Probabilistic language is two-sided by nature: "at least one head" has a twin, "all tails", and whichever probability you compute, the other should be one minus it. If A is a legitimate question, then "not A" is too, or the model lets you ask about winning but not about losing. Break the clause deliberately: take a coin model whose event family contains {head} but not its complement {tail} — you may price "heads" but are forbidden to price "tails". Arbitrage in the modeling layer itself.
Recall the first-head experiment from 0.4: the event "a head eventually appears" is A₁ ∪ A₂ ∪ A₃ ∪ … with An = "first head at toss n". If events are closed only under finite unions, every partial union A₁ ∪ … ∪ AN is a fine event — "the first head happens by toss N" — but the sentence you actually want to price is not an event. The clause guarantees: if each An can be priced, so can "at least one of them happens", over any number of them.
(i) Countable intersections. Complements turn unions into intersections (De Morgan's laws: (⋃ An)c = ⋂ Anc, and dually) — so a σ-algebra is automatically closed under countable intersections. "The average converges" is an intersection of union-of-tail events; the clause pair makes it a legitimate event before we ever compute it. (ii) Set differences. A ∖ B = A ∩ Bc ("in A but not B") is measurable whenever A and B are. Conditional thinking — "restrict attention to where B happened" — leans on this constantly.
1.1.3A bestiary: the smallest and largest σ-algebras
The smallest possible choice on any Ω: only two events. Model: a die is rolled, but the observer is in another room and hears only "something happened". Every non-trivial question — "was it even?", "was it a 6?" — is not an event, because the observer's information does not resolve it. Legitimate, minimal, and nearly useless — but it will return in 1.1.6 and 4.3 as the model of "no information".
If Ω is finite — dice, cards, urns — take F to be the power set 2Ω: the collection of all subsets, notation read "the set of subsets of Ω". Check the clauses: trivially true. This is exactly what 101 did silently, and for finite Ω it is perfectly safe — Vitali-style monsters need infinite structure to exist. Rule of thumb: on countable sample spaces, we almost always take all subsets as events (2.3 makes "almost" precise); on uncountable spaces, the choice of F becomes a real decision with real trade-offs.
Split Ω into disjoint pieces covering everything — a partition — and let F be all unions of pieces (including the empty union, ∅). A die is rolled in the next room; a friend reports only "low" ({1,2,3}) or "high" ({4,5,6}). Events the friend can resolve: ∅, {1,2,3}, {4,5,6}, Ω — a σ-algebra with four members. The friend cannot resolve "was it a 6?", because {6} splits no piece: it is not in F. Same outcome space as full 101 dice, coarser events — the first hint that F encodes information, not just set theory.
Let F consist of the sets that are countable (finite, or listable like {1, 2, 3, …} or the rationals) together with those whose complement is countable. Check: complements swap the two families; a countable union of countable sets is countable; a union of cocountable sets is cocountable (each already misses only countably much); a union mixing one cocountable set with anything is cocountable. All clauses hold: this is a genuine σ-algebra on the uncountable space Ω = ℝ — proof that on uncountable spaces there are meaningful choices other than "everything" or "almost nothing". But it is a poor home for the spinner: intervals like [0,½] are neither countable nor cocountable, so length is not even definable on it. Choice of F is a decision, and decisions can be wrong for the model at hand.
1.1.4What σ-algebras are not
The definition is easiest to internalize by breaking it. Each candidate below is a natural-looking collection of subsets that fails to be a σ-algebra — and each failure is one clause, in the wild.
Take F = all intervals [a,b] inside [0,1]. Tempting for the spinner: these are the sets we want to measure. But: the complement of [0, 0.3] in [0,1] is (0.3, 1] — not a closed interval, so clause 2 fails. And [0, 0.3] ∪ [0.5, 0.6] is not an interval at all, so clause 3 fails. The events we can price on the spinner are far more general than intervals — but, as 1.1.5 shows, they are the closure of the intervals, which is why the intervals are still the right place to start.
Upgrade to all finite unions of intervals. Complements now work (the complement of a finite union of intervals is another such union). The killer is clause 3 in its full countable strength: each singleton {0.5 + 1/n} is a degenerate interval, but the countable union {0.5 + 1/n : n = 1, 2, 3, …} is not a finite union of intervals, and adding its limit point to form {0.5} ∪ {0.5 + 1/n} needs a countable operation again. Geometry accumulates limit points; a domain closed only under finite operations drops them. This is exactly the "infinitely often" gap from 0.4.
All open sets: closed under finite unions and finite intersections, and every open set is a countable union of intervals, which looks promising. But complements fail: [0.2, 0.5] is closed, not open, yet it is the complement of an open set. Add the closed sets, then the sets that are open at some points and closed at others, then… the closure process only stops at the σ-algebra generated by the open sets — the Borel σ-algebra ℬ([0,1]), the true home of the spinner's events, constructed in 1.3.
On Ω = {1,2,3} take F = {Ω, {1}, {2,3}}. This one actually is a σ-algebra (check all three clauses — it is the partition σ-algebra of the split {1} | {2,3}). Now delete {2,3}: {Ω, {1}} fails clause 2, since {1}c = {2,3} is missing. σ-algebras are fragile under small edits: the clauses bind the collection together as a package.
1.1.5Generating a σ-algebra: the smallest good enough
You will rarely specify a σ-algebra by listing it — on uncountable spaces that is literally impossible to do explicitly. The standard move is: name some sets you insist on being able to measure, then take the smallest σ-algebra containing them.
For a collection 𝒞 of subsets of Ω (any collection at all), the σ-algebra generated by 𝒞, written σ(𝒞) and read "sigma of C", is the intersection of all σ-algebras on Ω that contain 𝒞. If 𝒞 is a family of preimages we overload the notation happily: for a random variable X, σ(X) is the σ-algebra generated by the sets {ω : X(ω) ≤ a} — 4.3.
Two things must be checked for this definition to make sense, and both checks are one-liners worth seeing (full proofs in 1.1.7): the intersection of σ-algebras is a σ-algebra, so σ(𝒞) is what it claims; and there is at least one σ-algebra containing 𝒞, namely the full power set, so the intersection is over a non-empty family.
(i) On the coin space Ω = {H, T}: σ({H}) = {∅, {H}, {T}, Ω} — the power set, because clause 2 forces {T} in, and clause 1 forces the rest. (ii) On a die, σ({6}) has exactly 4 events: ∅, {6}, {1,2,3,4,5}, Ω — the partition σ-algebra of "six or not-six", the natural σ-algebra of a game that only cares about rolling a 6. (iii) On [0,1], generating with all intervals produces the Borel σ-algebra: every set you can describe with countably many inequalities, comparisons, unions and intersections — the universe of the spinner.
Try to build a Vitali delegate set from intervals using countably many legal operations (complements, unions, intersections): you cannot. More strongly, in 1.3 we will see that the Borel σ-algebra — everything the operations can reach — does not contain it, and Chapter 3.4 will show the Borel sets are exactly the right domain: large enough that every experiment-relevant set is in it, small enough that every member can carry a consistent length. Generation is the honest way to say "we take exactly what the operations can reach, and no more" — no smuggled mystery sets, no delegates.
A π-system is a family closed under finite intersections (like the intervals of [0,1]); a λ-system (Dynkin system) is a family with Ω, closed under complements and under countable unions of increasing sequences. Dynkin's π–λ theorem (statement in 1.4) says: a σ-algebra is generated by any π-system whose generated σ-algebra it matches. This is the technical heart of "probabilities are determined by their values on nice sets" — why knowing a measure on intervals pins it down everywhere.
1.1.6σ-algebras as information
The partition example hid the idea that makes σ-algebras feel like more than bookkeeping. Suppose the die is rolled in the next room and a messenger reports to you — but the messenger's reports are limited. Every σ-algebra on a finite space is exactly the story of some reporting scheme:
On a finite space this picture is complete: every σ-algebra is the set of unions of the pieces of some partition, so "choosing F" and "choosing what the observer learns" are the same act. On uncountable spaces the slogan survives as the right intuition even though partitions become too crude to describe everything — the Borel σ-algebra is not a partition σ-algebra, but it is still best thought of as "what interval-reports can distinguish".
One trader sees every tick of the price process; her natural σ-algebra is fine enough to include each individual price-path event. Another sees only end-of-day closing prices; his σ-algebra is coarser, and the event "the price exceeded 100 intraday" is not an event for him — his information cannot resolve it, so no strategy of his can depend on it. Same Ω, different F, and the difference is the difference in information. When 4.3 defines σ(X) and Chapter 8 conditions on σ-algebras, this picture is the entire intuition.
On the die space with F = {∅, Ω}, try to price the event "6". You cannot: it is not an event, so the question is not merely unanswerable, it is malformed in the model. A model whose σ-algebra is coarser than the payoff structure cannot price that payoff — a modeling error, not a numerical one. In practice this is the error of conditioning on information you do not have; it returns with force when we meet conditional expectation.
1.1.7Two honest proofs, and what they teach
Two claims above were deferred to here. Both proofs are short, and both teach a pattern this course will reuse constantly: closure clauses combine.
Let (Fi)i∈I be any family of σ-algebras on Ω (the index set I may be uncountable). Then G = ⋂i∈I Fi is a σ-algebra.
Clause 1: Ω is in every Fi — that is clause 1 for each of them — so Ω ∈ G; likewise ∅. Clause 2: take A ∈ G. Then A ∈ Fi for every i; each Fi being a σ-algebra, Ac ∈ Fi for every i; hence Ac ∈ G. Clause 3: take a sequence A₁, A₂, … ∈ G. For every i, all the An lie in Fi, so ⋃n An ∈ Fi — clause 3 for that particular Fi. This holds for every i, so ⋃n An ∈ G.
What the proof teaches: to verify a property of an intersection, verify it "member by member". This is the exact pattern that makes σ(𝒞) well-defined and, later, the pattern behind monotone class arguments: complicated objects are checked by decomposing into simpler ones that each already pass the test.
If F is a σ-algebra and A₁, A₂, A₃, … ∈ F, then ⋂n An ∈ F.
By De Morgan's law, ⋂n An = (⋃n Anc)c. Check the right-hand side is in F: each Anc ∈ F by clause 2; their union is in F by clause 3; its complement is in F by clause 2 again. Three clauses, three moves.
Why this little fact matters enormously: "the average converges" is the event that the running average eventually stays within ±ε of its limit — a countable intersection (over ε = 1, 1/2, 1/3, …) of countable unions (over starting points) of simpler events. Proposition 2 is what guarantees the Law of Large Numbers is a statement about an event at all. Small clause, big consequences — a theme of the whole course.
We can now say precisely what a probability model's domain is. What is missing: the numbers. In 1.2 we build generated σ-algebras in earnest, in 1.3 we construct the Borel σ-algebra ℬ(ℝ) — the single most important σ-algebra in the course, because it is the one every distribution function silently lives on — and in 1.4 the π–λ machinery that makes measures determined by their values on intervals.