Generated σ-Algebras: The Smallest Good Enough
Nobody ever writes down a σ-algebra. You declare a handful of events you care about, and the rules force the rest into existence — this subchapter is about exactly what "the rest" means.
In this subchapter
1.2.1The problem: too many events to name
On a finite sample space, life is easy: if Ω has n outcomes, the full power set P(Ω) — the collection of all subsets — has 2n members, and we saw in 1.1 that it is a perfectly good σ-algebra. Declare every subset an event, assign each outcome a nonnegative mass, add to one, done.
On a serious sample space both comforts die. Take Ω = ℝ: the collection of all subsets is unimaginably vast (we will count it in 1.3), nobody could list it, and the Vitali construction from Chapter 0 says that no sane probability can be defined on all of it anyway. Or take an infinite sequence of coin flips: the events we actually care about — "the first head arrives by toss n", "the long-run frequency of heads exceeds 1/2" — are built from a few simple declarations by unions, intersections, complements, and limits.
So the honest workflow is the reverse of what 101 suggests. We do not choose a σ-algebra; we declare a family of basic events 𝒜 — the questions we intend to ask — and then take whatever the σ-algebra rules force upon us. That forced collection is written σ(𝒜) and called the σ-algebra generated by 𝒜. This subchapter makes that idea precise, computes it in small cases, and — just as important — clears up what it does not mean.
1.2.2Intersections behave: the smallest σ-algebra above a family
The whole construction rests on one humble observation. To intersect collections (not sets: collections of sets), recall the language: ⋂i Fi is the collection of sets that belong to every Fi. So if each Fi is a σ-algebra on Ω, a set in the intersection passes every test in every Fi.
Let Fi, i ∈ I (any index set whatsoever — one, two, countably, or uncountably many), be σ-algebras on the same Ω. Then ⋂i∈I Fi is a σ-algebra on Ω.
Check the three clauses from 1.1. If A is in the intersection, it is in each Fi, so Ac is in each Fi, so Ac is in the intersection. Likewise for a countable union: if A₁, A₂, … all lie in the intersection, each Fi contains them all, so each contains their union, so the union is in the intersection. And Ω is in every Fi. The proof is a three-line mirror game: the clauses are universally quantified statements, and universal statements survive intersection. ∎
Note what makes this work: each clause of the definition is preserved under taking sets that satisfy it in every member. Contrast with unions: a union of σ-algebras is generally not a σ-algebra — that failure is exactly why generation is an operation at all.
For any family 𝒜 of subsets of Ω (no structure assumed on 𝒜 at all), set
the intersection of all σ-algebras containing 𝒜. The collection being intersected is nonempty — P(Ω) is one of its members — so by Theorem 1, σ(𝒜) is a σ-algebra, it contains 𝒜, and every σ-algebra that contains 𝒜 also contains σ(𝒜). In words: σ(𝒜) is the smallest σ-algebra containing 𝒜.
Two sanity checks, computed completely.
σ(∅) — the family 𝒜 is empty, so we are asking for the smallest σ-algebra on Ω, full stop. Every σ-algebra must contain ∅ and Ω, and {∅, Ω} is itself a σ-algebra (check the clauses: complements of ∅ and Ω are each other; unions of its members stay inside it). So σ(∅) = {∅, Ω}: the "no questions asked" σ-algebra, on which no experiment can distinguish anything.
σ({A}) for a single event A. Any σ-algebra containing A must contain Ac, ∅, and Ω. And {∅, A, Ac, Ω} is a σ-algebra — the clauses close up over it (a union of members is either one of the members or Ω). So σ({A}) = {∅, A, Ac, Ω}: four events, exactly the questions "did A happen, did it not, everything, nothing". Ask one question, get four events. The rules do the rest.
1.2.3Two events: atoms and the sixteen
Two events A, B is the smallest case where something interesting happens, and its structure is worth seeing once in full because it scales into the central definition of the next chapters.
The atoms generated by A, B are the four sets built by choosing, for each of A and B, whether the point is in it or its complement:
Each atom is in σ(A, B) (σ-algebras are closed under the two operations used to build them). The atoms are pairwise disjoint, and their union is Ω — a partition: every outcome lands in exactly one atom.
Let A₁, A₂, A₃, A₄ be the four atoms of A, B (some may be empty). Then σ(A, B) consists of all unions of sub-collections of atoms: at most 2⁴ = 16 events.
Two inclusions. The family of all unions of atoms is a σ-algebra: complements are unions of the complementary atoms, and a countable union of unions of atoms is a union of atoms — the atoms act as indivisible building blocks. It contains A = (A∩B) ∪ (A∩Bc) and similarly B, so σ(A, B) is contained in it, being the smallest. Conversely, every atom is in σ(A, B) by closure under ∩ and c (Proposition 2 of 1.1), so any union of atoms is too. Both walls meet. ∎
Ω = {HH, HT, TH, TT}, A = "first flip heads" = {HH, HT}, B = "second flip heads" = {HH, TH}. The atoms: {HH}, {HT}, {TH}, {TT} — the atoms are singletons here, so σ(A, B) = P(Ω), all 16 subsets. The two questions together determine everything; each alone (Example 2 structure) would have left us blind to the joint behavior.
The pattern generalizes: n events cut Ω into at most 2n atoms, and the generated σ-algebra has at most 22n events. Finite families generate finite σ-algebras. The interesting question is what happens with infinite families — and the answer is subtle enough to deserve its own section.
1.2.4Countable generators: two very different outcomes
Start with the friendliest infinite case.
Let Ω = ℕ = {1, 2, 3, …} and let 𝒜 be the singletons: {1}, {2}, {3}, …. Then σ(𝒜) = P(ℕ). Why: any subset S ⊆ ℕ, being a subset of a countable set, is itself countable — listable as s₁, s₂, s₃, … — so S = ⋃k {sk}, a countable union of singletons. Every subset of ℕ is therefore reachable, and countable additivity is exactly the closure property that reaches it. This is why discrete probability "just works": on a countable space, singletons generate everything, and we may assign masses to singletons without ever worrying about the domain.
Now run what looks like the same construction on Ω = ℝ, and watch it fail to produce anything like the sets a continuous model needs.
Take Ω = ℝ and again the singletons {x}, x ∈ ℝ. What is σ(singletons)? Certainly not P(ℝ) — and in fact the interval (0, 1) is not in it. The reason: a countable union of singletons is a countable set; complements of countable sets are co-countable (their complement is countable); and the family of countable and co-countable sets is already a σ-algebra containing every singleton. By minimality, σ(singletons) is exactly the countable–co-countable σ-algebra. The interval (0, 1) is neither countable nor co-countable — it is not on the list. Lesson: the shape of the generator is not enough; the ambient space decides what the generator can reach. To capture intervals on ℝ we must put intervals in the generator — that construction, the Borel σ-algebra, is 1.3.
Example 4 and Example 5 use the same generator (singletons) on two spaces and land in utterly different places: everything on ℕ, almost nothing on ℝ. When someone says "the σ-algebra generated by X", always ask: generated on which space, by which declared questions? The phrase is meaningless without both.
1.2.5What generated does not mean: closure is not a recipe
It is tempting to read σ(𝒜) operationally: "start with 𝒜, apply complements and countable unions, repeat until nothing new appears, and list what you got." The first half is right; the second half is a trap, and an instructive one.
Iterating the operations does climb the wall: stage 0 is 𝒜, stage n + 1 adds complements and countable unions of things built by stage n. But there are sets in σ(𝒜) that appear at no finite stage, and — the honest punchline — even taking the union of all countably many stages is not enough in general: one must keep climbing through transfinite stages, and for the Borel σ-algebra genuinely uncountably many stages are needed. This is not a pathology of a weird example; it is the standard situation. We flag it and move on: for using generated σ-algebras, the smallest-closed-family definition is all we ever need — but never promise yourself a listing.
What is true, and is the working tool throughout the course, is a one-way principle:
If F is a σ-algebra with 𝒜 ⊆ F, then σ(𝒜) ⊆ F. To prove that some set E belongs to σ(𝒜), it suffices to exhibit any σ-algebra containing 𝒜 that contains E.
σ(𝒜) was defined as the intersection of all σ-algebras containing 𝒜; F is one of the collections being intersected, so the intersection is inside it. ∎
It sounds too easy to be useful. It is used on every page of the rest of the course: to show {X ≤ t}-type sets are measurable, to show a limit of measurable functions is measurable, to show the "eventually" and "infinitely often" events exist. Each time, we quietly find a σ-algebra that already contains the set — and minimality does the work.
1.2.6Why this matters: infinite coin flips and cylinder sets
Here is the payoff example, and the model for every stochastic process. Let Ω = {H, T}∞ — an infinite sequence of coin flips, an outcome being the entire forever-list. For a finite word w of length n, the cylinder set Cw is the event that the first n flips agree with w. Declare the cylinders as the basic questions. What do the rules force into σ(cylinders)?
The event that heads appears at least once is ⋃n≥1 {flip n is H} — a countable union, and each {flip n is H} is a finite union of cylinders (one per word of length n − 1). So it is in the generated σ-algebra. Similarly "heads infinitely often" — the event of Chapter 0 that no finite additivity could weigh — is
an intersection of unions of events in the family: the closure clauses are doing exactly the work that the failure in Chapter 0 demanded.
The set of sequences that are eventually constant — all heads from some point on, or all tails — is in the σ-algebra (a countable union over m of the events "constant from m on"). But consider the event "the sequence, read as a binary expansion, lands in the Vitali delegate set V ⊆ [0, 1]" from 0.3: mapping sequences to [0, 1] and asking about V, we get a set that no length-like measure can weigh — a genuine event-shaped object that the generated σ-algebra of nice cylinders cannot honestly host. Generation gives us the sets the rules force; it cannot rescue sets the rules were never meant to weigh. Chapter 3 draws this boundary precisely.
Take stock. A σ-algebra is not a container you pick off a shelf; it is the closure of the questions you commit to asking. Ask about individual outcomes — you get the power set on countable spaces, and something much thinner on uncountable ones. Ask about intervals — you get the Borel σ-algebra, the subject of 1.3. Ask about cylinders — you get the natural habitat of stochastic processes. The generator is the modeler's actual act of modeling.
You never choose a σ-algebra. You declare the questions you intend to ask, and the rules force the rest into existence.
We can now generate σ-algebras from declared questions and know what minimality buys us. Next: the single most important generated σ-algebra in probability — ℬ(ℝ), generated by the intervals — and the discovery that a countable family of declared questions suffices to generate it, which is the seed of the uniqueness theorems in 1.4.