The Borel σ-Algebra on ℝ
Every random variable reports real numbers, so every event worth asking about is a set of reals. The Borel σ-algebra is the answer to: which sets of reals deserve probabilities?
In this subchapter
1.3.1Why the real line needs its own σ-algebra
A random variable — a price, a return, a duration, a temperature — is a number. Ask any practical question about one and you get an event of the shape {ω : X(ω) ≤ t}: "the price is under 100", "the loss exceeds the VaR line". Since Chapter 4 will make precise how such sets inherit measurability, the burning question of this subchapter is the target: on Ω = ℝ itself, which subsets get to be events?
Three candidate answers, and why each fails. The full power set P(ℝ): impossible — the Vitali failure of 0.3 lives exactly here. The singletons and what they generate, from 1.2: too thin — Example 5 there showed the interval (0, 1) is not even in it. Some hand-curated list of "reasonable" sets: unprincipled — the moment your theorem needs one more set, the list is stale. What we want is a σ-algebra that contains everything a ruler or a limit could ever carve out of the line, and nothing it could not. That is the Borel σ-algebra.
The Borel σ-algebra on ℝ is the σ-algebra generated by the open sets:
Its members are called Borel sets. (Recall from 1.2: the right-hand side is the smallest σ-algebra containing every open set — equivalently, the intersection of all σ-algebras that contain them.)
"Open set" is analysis vocabulary, so a one-line gloss: a set U is open if every point of U is surrounded by a little interval still inside U — no boundary points included. Open intervals (a, b) are open; [a, b] is not (the endpoint a sits on the boundary). Intuitively: sets you can draw on the line without drawing their edges.
1.3.2The definition, through three equivalent doors
The definition uses open sets, but for a probabilist the workhorse generators are intervals and half-lines. The theorem below says all roads lead to the same ℬ. Its proof is the first place where density of the rationals — the fact that the rational numbers ℚ sit arbitrarily close to every real — does honest mathematical work, so watch for it.
All of the following generate the same σ-algebra on ℝ:
- all open sets;
- all open intervals (a, b);
- all half-lines (−∞, t];
- all intervals (a, b) with a, b rational.
By the minimality principle of 1.2, it suffices to show each family is contained in the σ-algebra generated by the previous one, all the way around.
(1) ⊆ (2): every open set is a countable union of open intervals. Proof sketch: let U be open. Each x ∈ U has an interval (x − ε, x + ε) ⊆ U; shrink it to an interval with rational endpoints still containing x and still inside U (possible because ℚ is dense — there is a rational between any two reals). There are only countably many rational-ended intervals in total, so we have written U as a union over a sub-collection of a countable family — countable. Then σ(intervals) contains every open set, so (1)'s σ-algebra is inside it.
(2) from (3): the half-line (−∞, t] is closed (its complement (t, ∞) is open), and in σ-algebra terms: (−∞, t] = ⋂n (−∞, t + 1/n) — an intersection of open half-lines, each of which is an open set — so half-lines are in σ(open sets). Conversely (a, b) = (−∞, b) ∖ (−∞, a] and (−∞, b) = ⋃n (−∞, b − 1/n], so intervals are in σ(half-lines). Both directions at once.
(2) from (4): any open interval (a, b) is the union of the rational-ended intervals inside it — for each x ∈ (a, b) pick rationals p, q with a < p < x < q < b, again by density — and a countable union of a sub-family of a countable family is countable. So the rational intervals, a countable family, generate everything the uncountable family (2) does.
The circle closes: all four families generate exactly ℬ(ℝ). ∎
A distribution function F(t) = P(X ≤ t) assigns probabilities to half-lines (−∞, t]. Theorem 1, door (3), says those half-lines already generate all of ℬ(ℝ). This is why "knowing the distribution function" will turn out to mean "knowing the probability of every Borel set" — the fact behind the uniqueness result in 1.4 and its payoff in 4.2.
1.3.3What is Borel: a working collection
A fast tour of what is in — each justified by the generator, not by fiat.
A singleton is Borel: {x} = ⋂n (x − 1/n, x + 1/n), a countable intersection of open intervals. Hence every countable set {x₁, x₂, …} is Borel (countable union), and so is every set whose complement is countable. Compare Example 5 of 1.2: the singletons were always going to be Borel; the point there was that they do not generate ℬ.
The Cantor set of 3.3 is closed (it is an intersection of closed sets, each a finite union of closed intervals), and closed sets have open complements, so every closed set — including the Cantor set, including every finite union of intervals of any flavor — is Borel. The set we will build in 3.3 to be uncountable yet invisible to length is, throughout, a perfectly good event.
Let f₁, f₂, … be continuous real functions, and ask: at which inputs x does the sequence fn(x) converge? The Cauchy criterion, run over rational tolerances, writes this set as
Each innermost set is closed (inverse image of a closed interval under a continuous function), and the whole expression is a countable sandwich of unions and intersections. So "the set of points where the algorithm converges" is Borel — automatically, no case-by-case check. This is the pattern of the whole course: closure clauses turn practical questions into events.
1.3.4What is not Borel: most sets, actually
How big is ℬ(ℝ)? The honest answer requires counting beyond the finite — a two-line detour with a gloss. Cardinality: two collections have the same size if their members can be paired off one-to-one. The reals have the size of the continuum, written 𝔠 (the same size as the collection of all infinite binary sequences). Cantor showed there are strictly more sub-collections than members: the subsets of ℝ have size 2𝔠 > 𝔠 — the same diagonal move as in the Vitali preview.
There are 𝔠 many Borel sets, but 2𝔠 many subsets of ℝ. Consequently most subsets of ℝ are not Borel: if you could pick a subset of ℝ uniformly at random from all subsets, it would be non-Borel with certainty.
Upper bound: by Theorem 1 door (4), ℬ is generated by a countable family of intervals. A careful bookkeeping of the countable operations (the transfinite climb flagged in 1.2 turns out to stop at a countable collection of stages, each adding at most 𝔠 sets) gives at most 𝔠 Borel sets. Lower bound: the intervals alone are 𝔠 many. Both bounds proved in full in any set theory text; we take the upper bound as flagged and use only the conclusion. ∎
Here is the trap: the proposition says non-Borel sets are everywhere, but it does not name one — every set you can write down with intervals, limits, and standard operations is Borel by construction (Example 4 is the pattern). Producing a specific non-Borel set requires an act of choice: the Vitali delegate set V from 0.3 is the classical example, and in 3.4 we will show V cannot be Borel — indeed cannot carry any length-like probability at all. The lesson for practice: Borel is big enough for anything you can describe, small enough to stay measurable. The pathologies live in the undescribable majority.
1.3.5The rational generator: a countable family suffices
Underline the fact that will power 1.4. From Theorem 1, door (4):
ℬ(ℝ) = σ(ℐ) where ℐ is the countable family of intervals with rational endpoints. One declared question per rational pair, countably many in total, forces every Borel set into existence. A measure will only ever need to be specified on a countable family to be pinned down everywhere — the exact statement is the uniqueness theorem of 1.4.
Contrast the countable generator with a natural uncountable one: all singletons {x}, one per real. From 1.2 Example 5, that uncountable family generates only the countable–co-countable σ-algebra — far smaller than ℬ — while the countable family of rational intervals generates everything. More generators is not more power: the shape of the questions, not their number, is what matters. Probabilistically: asking "is X in (a, b)?" for rational a < b is an informationally complete interrogation; asking "is X = x?" for every real x is not.
1.3.6Beyond the line: products and previews
The construction transports. On ℝ², declare the rectangles (a, b) × (c, d) and take the generated σ-algebra: that is ℬ(ℝ²), the home of joint distributions and of Fubini's theorem in Chapter 6. On [0, 1], take the intervals inside the unit interval: the same construction gives ℬ([0,1]), the domain where Lebesgue measure will live in 3.2. On the space of infinite coin flips, the cylinders of 1.2 play exactly the role the intervals play here. One pattern, every space: declare the coordinate questions, close under the rules.
The Borel σ-algebra is the answer to a modeling question: which sets of reals can any ruler, any limit, or any loss function ever carve out? Exactly the Borel ones.
We have the domain on which probabilities on the line will live, and the key structural fact that a countable family of declared questions generates it. What we lack is the numbers: if two candidate probability measures agree on intervals — two quants agree on every threshold risk — must they agree everywhere? Yes, and the reason is a structural theorem about families of sets: π-systems and Dynkin systems, next.