Information as a σ-Algebra: σ(X) and Doob–Dynkin
"X is measurable with respect to F" has been read as a license. It has a second reading: "F contains at least the information to determine X". This subchapter makes that reading exact, and proves the theorem — Doob–Dynkin — that turns it from metaphor into machinery.
In this subchapter
4.3.1The generated σ-algebra, again and deliberately
The construction is a reunion of two ideas already met separately: generation (1.3) and measurability (4.1). Intersect the two:
For a random variable X: Ω → ℝ, let
Two descriptions, one object: the collection of all inverse images of Borel sets (a σ-algebra by the commutation facts of 4.1 Theorem 1's proof), and the intersection of all σ-algebras on Ω with respect to which X is measurable (nonempty: the full power set qualifies; intersections of σ-algebras are σ-algebras — 1.2). The second description is the definition of σ(generated) in general: σ(any family of sets or functions) is the smallest σ-algebra containing them / making them measurable.
A die roll (Ω = {1,…,6}): σ(X) for the identity is the full power set (every set is X−1(B) for suitable B); σ(X) for "X = 1 if even, 0 if odd" is the four-element σ-algebra {∅, {2,4,6}, {1,3,5}, Ω} — coarse, and exactly as coarse as the variable deserves. On [0,1]: σ(indicator of the Cantor set) is {∅, C, Cc, Ω} intersected appropriately with the ambient story; σ(identity on ([0,1], ℒ)) is all of ℒ, since inverse images of Borel sets under the identity are the Borel sets themselves, and Borel ⊂ ℒ — but note the two σ-algebras are different: σ(identity) = ℬ([0,1]) when the space is ([0,1], ℬ), a strict sub-family of ℒ. The variable's precision, not the ambient space's richness, sets the size of σ(X).
4.3.2σ(X) as information: the reading test
Here is the reading that changes everything downstream. Think of a σ-algebra not as "a collection of legal sets" but as an information state: the answers to all the yes/no questions some observer can settle. Then:
- G ⊆ F ("G is a sub-σ-algebra"): the G-observer is coarser — knows less; F's questions may be unanswerable for G.
- X is G-measurable: the G-observer can determine X's value exactly — X's question-set {X ≤ t} lies in G for all t, so the observer who can settle G can settle X.
- σ(X): exactly the information content of observing X — no more (every set in σ(X) is settled by knowing X's value: it is X−1(B), determined by which Borel band X fell into) and no less (any observer who can determine X can settle every {X ≤ t}, hence all of σ(X)).
This is why Example 3 of 4.1 felt like a paradox: the identity on [0,1] fails measurability against the four-set σ-algebra because the four-set observer cannot determine a real number to full precision — the observer knows only which half of [0,1] the outcome fell in.
Fix the week's outcome space Ω (all price paths) and let G = σ(closing price on Monday). A G-observer knows Monday's close and nothing else. Variables measurable with respect to G: Monday's close itself, its sign, any function of it. Not measurable: Tuesday's close, the week's maximum, Monday's intraday path. If H = σ(Monday close, Tuesday close), then G ⊆ H — the two-day observer strictly refines the one-day observer, and the inclusion of σ-algebras is the passage of time. §5 formalizes this.
4.3.3Doob–Dynkin: functions of X, and only those
The dictionary makes a claim that deserves proof: that "measurable with respect to σ(X)" and "is a function of X" are the same thing. The claim is true, and it is the hinge of the chapter.
Let X, Y: Ω → ℝ. Then Y is σ(X)-measurable if and only if there exists a Borel function g: ℝ → ℝ with Y = g(X) — that is, Y(ω) = g(X(ω)) for every ω.
(⇐) is the easy arrow: if Y = g(X) with g Borel, then {Y ≤ t} = X−1(g−1((−∞,t])); the inner set is Borel (g is Borel), so the level set is in σ(X) by definition. Composition of measurables is measurable — the rule from 4.1 Example 6, now on the books.
(⇒) Climb the ladder. If Y = 1A with A ∈ σ(X): then A = X−1(B) for some Borel B (definition of σ(X)), so Y = 1B(X) — take g = 1B. If Y is a simple function — a finite sum Σ ak 1Ak with Ak ∈ σ(X) — the same representation term by term gives Y = g(X) for g = Σ ak 1Bk, Borel. For general σ(X)-measurable Y: approximate by simple functions — define Yn by cutting the range into bands of height 1/n (the construction of 5.1 previews here; banding is the same trick) — each Yn is σ(X)-measurable and simple, so Yn = gn(X); and Yn → Y pointwise, since the bands shrink. Define g(x) := limsupn gn(x) where the limit exists, 0 elsewhere — a Borel function (limsup of Borel functions is Borel: 4.1 Theorem 2), and g(X(ω)) = Y(ω) at every ω. ∎
If Y is σ(X)-measurable, then Y is a function of X — you can compute Y from the single number X(ω) with a Borel rule. Everyday version: "the P&L of an option position at expiry is a function of the underlying's price" is the statement "P&L is σ(price at expiry)-measurable", and Doob–Dynkin upgrades the financial shorthand to an existence theorem: there is a genuine measurable payoff function. The theorem also disciplines overreach: the function g lives on ℝ — it may be wild (any Borel function), but it consumes only the value of X, not ω. If your derivation of Y ever "peeks" at the outcome beyond X, you have left σ(X)-measurability behind, whether or not you intended to.
On ([0,1], ℒ, λ) let X = 1[0,1/2] and Y = 1[0,1/4]. Y is not σ(X)-measurable: σ(X) = {∅, [0,1/2], (1/2,1], Ω} does not contain {Y = 1} = [0,1/4]. Correspondingly no function g can reproduce Y from X — X reports only "left half or right half", and no rule on two values can split the left half further. The obstacle is not analytic (Y is a nice indicator) but informational: X's resolution is too coarse to determine Y. Contrast: Y is measurable with respect to σ(Y) ∨ σ(X) (the smallest σ-algebra containing both — note the join notation ∨, standard for "coarsest common refinement"), and with respect to any σ(X, Y) refinement. "Measurable with respect to" is a partial order on information; measurability questions are ordering questions.
4.3.4Coarse versus fine: the hierarchy in one variable
The smallest σ-algebra {∅, Ω} carries exactly one real-valued measurable variable: constants (check the level-set test: {c ≤ t} is ∅ or Ω). The observer with zero information can determine only quantities that never vary. This is the anchor of the hierarchy — and the reason conditional expectation (8.1) will, given zero information, reduce to the ordinary expectation: conditioning on knowing nothing must return the unconditional answer.
4.3.5Filtrations in miniature: information arriving over time
A filtration is an increasing sequence of σ-algebras F₁ ⊆ F₂ ⊆ F₃ ⊆ ⋯ ⊆ F — one information state per time step, each containing everything the earlier ones knew. A sequence of variables X₁, X₂, … is adapted to the filtration if Xn is Fn-measurable for each n: at time n, the value Xn is already determined by what is known.
Ω = all coin-flip sequences, Fn = σ(first n flips): F₁ distinguishes 2 classes of paths, F₂ distinguishes 4, Fn 2n — each Fn is generated by a finite partition (the atoms of 2.3 here doing honest work as information cells), and ∪Fn generates a σ-algebra in which "the limiting fraction of heads" is measurable — it is a pointwise limit of Fn-measurable averages, and Theorem 2 of 4.1 closes the countable limit. Everything in the strong law (7.2) and in martingale theory (8.3) lives on exactly this scaffolding: a filtration is the course's model of time.
4.3.6Why the course needs this: the view ahead
Chapter 4 assembled: measurability as credential and as information (4.1, here), laws as CDFs (4.2), and functions-of-observables as a theorem (Doob–Dynkin). The architecture is now complete enough to support the weight of the second half:
- Expectation (Chapter 5) is defined for σ(X)-measurable objects and computed via laws; Doob–Dynkin is what makes "expectation of g(X) depends only on the law of X" a theorem rather than a hope.
- Independence (Chapter 6) is a statement about σ-algebras; variables are independent exactly when their σ-algebras interact trivially. Fubini will be proved on products of σ-algebras.
- Convergence (Chapter 7) distinguishes modes precisely because they ask different questions of the same filtration hierarchy.
- Conditioning (Chapter 8) — the payoff — is defined as follows: E[Y | G] is the G-measurable variable that does Y's best approximation "on average"; the Doob–Dynkin lemma is what makes "the conditional expectation given X" a function of X, hence writable as E[Y | X] = g(X) with a genuine g. Conditioning is Doob–Dynkin's second act.
σ(X) is what X knows. Doob–Dynkin says: what X knows is exactly what can be computed from X — and nothing else gets to call itself a function of X.
The objects are built; numbers come next. Chapter 5 assigns every random variable its number — the expectation — built from simple functions upward, with the three convergence theorems (monotone, Fatou, dominated) that make integrals safe to take limits in. The first subchapter starts where this one's proof already visited: banding the range, simple functions, and the first honest integral.