Independence as Product Measure
"Independent" is the most abused word in applied probability. This subchapter gives it a definition with teeth — events that split, σ-algebras that split, measures that multiply — and the standard warning that the naive reading of the word is a trap.
In this subchapter
6.1.1Independence of events: the familiar case, sharpened
Events A, B are independent if P(A ∩ B) = P(A)P(B). A family {Ai} (finite or infinite) is mutually independent if every finite subfamily multiplies:
for every choice of distinct indices. The "every finite subfamily" clause is the teeth: independence is a property of the whole family, checked on all its corners at once.
Two fair dice, Ω = {1,…,6}² with uniform mass 1/36: A = "first die is even" has P = 1/2 (18 of 36 outcomes); B = "second die shows ≥ 5" has P = 1/3 (12 of 36); A ∩ B has 6 outcomes ({2,4,6} × {5,6}), so P(A ∩ B) = 6/36 = 1/6 = (1/2)(1/3). Independent — and the verification was counting, not faith. The same Ω supplies the contrast cases. Let C = "sum is even": P(C) = 1/2 (both even or both odd, 18/36) and P(A ∩ C) = 9/36 = 1/4 = P(A)P(C) — independent, for the quiet reason that parity of the sum cares about the two dice symmetrically. Let D = "sum ≥ 9": P(D) = 10/36, but P(A ∩ D) = 3/36 ≠ 5/36 = P(A)P(D) — Dependent: knowing the first die is even rules out the high-odd sums. The lesson is procedural: independence is a multiplication check, run on explicit events, never an eyeball judgment.
The most common practitioner slip: "these risks are independent because they don't overlap." Two disjoint events with positive probabilities are maximally dependent: P(A ∩ B) = 0 ≠ P(A)P(B) > 0, and knowing A occurred tells you B certainly did not. Overlap-in-Ω and probabilistic independence are unrelated concepts — the correct mental image is a rectangle-area check (P(A)P(B) is the area of the rectangle in the unit square; independence says the actual overlap region has exactly that area), not a Venn-diagram glance.
6.1.2Independence of σ-algebras and variables
Sub-σ-algebras F₁, F₂, … are independent if every choice of events Ai ∈ Fi multiplies. Random variables X₁, X₂, … are independent if their generated σ-algebras σ(X₁), σ(X₂), … are — i.e. every selection of level-set questions multiplies. Events are independent iff their generated σ-algebras are (check: σ(A) = {∅, A, Ac, Ω}, four multiplication checks, three trivial).
Defining independence of variables through σ-algebras looks heavy, but it is the only version that composes: the point of independence is that anything you compute from X is independent of anything you compute from Y. Doob–Dynkin (4.3) says "anything computed from X" is exactly "the σ(X)-measurable functions"; so defining independence at the σ-algebra level and using Doob–Dynkin yields the practical statement: if X, Y are independent and g, h are measurable, then g(X), h(Y) are independent — one line, and the entire reason the definition is shaped this way. (The event version alone could not deliver this: "g(X) ≤ t" is not an event "about X" in any pre-registered list.)
6.1.3The product measure: independence exists
So far, independence is a property a given family may or may not have. The construction theorem says the property is always achievable — there is a space on which any desired laws coexist independently. This is what licenses "let X, Y be independent with laws μ, ν" — a sentence practitioners utter daily and the theory must underwrite.
Given probability spaces (Ω₁, F₁, P₁) and (Ω₂, F₂, P₂), there is a unique probability measure P₁ ⊗ P₂ on the σ-algebra F₁ ⊗ F₂ := σ(A × B : A ∈ F₁, B ∈ F₂) (generated by rectangles) with
for every rectangle. Under P₁ ⊗ P₂, the coordinate maps X(ω₁, ω₂) = ω₁ and Y(ω₁, ω₂) = ω₂ are independent with laws P₁, P₂.
Existence. Define the would-be measure on the semiring of rectangles by multiplication (that is the formula). The hard part — that rectangle-multiplication extends to a countably additive measure on all of F₁ ⊗ F₂ — is the Carathéodory extension theorem again, in exactly the role it played for Lebesgue measure (3.2): a pre-measure on a generating family extends uniquely. We do not re-run the ε-management (the argument is verbatim the §4 machinery of 3.2 with rectangles replacing intervals); the course flags this as the same theorem, second application, and moves on. Uniqueness. The rectangles form a π-system generating F₁ ⊗ F₂; two probability measures agreeing on it agree everywhere — π–λ (1.4), one line. Independence of the coordinates. X is measurable (level sets: {X ∈ A} = A × Ω₂, a rectangle); σ(X) is exactly the rectangles with full second factor; for A ∈ σ(X), B ∈ σ(Y): P(A ∩ B) = P((A × Ω₂) ∩ (Ω₁ × B)) = P(A × B) = P₁(A)P₂(B) = P(A)P(B). ∎
n fair coin flips: Ω = {H,T}n with uniform mass — exactly ({H,T}, 2{H,T}, ½δ)n, the n-fold product; the flips are independent by construction, and every "the coins don't remember" argument of elementary probability is this theorem in disguise. The uniform pair (X,Y) on the unit square: ([0,1]², ℒ⊗ℒ, λ⊗λ) — λ⊗λ is "area" (3.1's outer measure on the plane, now constructed rather than asserted). Infinite sequences of iid draws — the object every Monte Carlo run pretends to be — are the countable product; the same extension theorem covers it, with the one new wrinkle that the space is now uncountable and the σ-algebra is generated by cylinder sets (all finitely many coordinates specified). This is the space the strong law (7.2) lives on.
6.1.4Pairwise is not mutual: the XOR example
Flip two fair coins: X, Y the two ±1-valued outcomes, and Z = XY (the XOR in ±1 clothing). Claim: X, Y, Z are pairwise independent but not mutually. Each pair: (X,Z) — P(X = 1, Z = 1) = P(Y = 1) = 1/2 = P(X=1)P(Z=1), and the other corners match by symmetry; similarly (Y,Z) and (X,Y) (the last is independence by construction). But the triple: P(X = Y = Z = 1) = P(Y = 1, X = 1) = 1/4 ≠ 1/8 = (1/2)³ — any two of the three determine the third, so the three-way corners fail. The definition's "every finite subfamily" clause is what this kills: pairwise is 3 checks, mutual is 4 (with the triple), and the missing check is where naive practice goes to die. Risk version: three desk exposures, each pair looking uncorrelated, jointly determined by two underlying factors — a "fully diversified" book that is nothing of the kind. Correlation-zero is weaker still than pairwise independence (6.2 has the summation example that separates those too); the ladder is strict at every rung.
A second structural warning: from independent X₁, X₂, …, functions of disjoint index sets remain independent (the σ-algebra argument extends: σ(X₁,…,Xn) and σ(Xn+1,…) are independent — the "tail" σ-algebra, which 6.3 revisits in its Borel–Cantelli role). But a variable like X̄n = (X₁+⋯+Xn)/n is independent of nothing downstream — it is a function of the whole prefix, and as n grows the prefixes overlap. Practical content: the belief that successive Monte Carlo averages "wash out dependence" is unlicensed; the correct tools for functions of overlapping independence are the limit theorems of Chapter 7, applied once, at the end — not iterated intuition.
6.1.5What independence buys
If X, Y are independent and both integrable (or both nonnegative), then E[XY] = E[X]E[Y]. Consequently: the variance of a sum of independent variables is the sum of variances (via E[XiXj] splitting for i ≠ j) — the computation under every VaR-aggregation formula; and the characteristic function of a sum is the product of characteristic functions — the tool that will prove the CLT (7.3). Proof of the factoring: run the standard machine (indicators → simple → nonnegative → general) — for indicators of level-set events it is the definition of independence; each rung uses only operations preserved by independence (Theorem of §2's note). We run it in full in 6.2 in its Fubini form, where it becomes one clean line.
6.1.6What independence does not mean
- Independent ≠ uncorrelated. Correlation zero is one equation about one pair of functions; independence is all equations about all functions. (Example 4's cousins separate them; 6.2 Example 4.)
- Independent ≠ "no causal link". Causation is outside the model entirely; independence is a multiplication property of a measure. Deterministic functions of independent variables are independent of their arguments' sources only when the function uses its own variable — a tautology worth restating: X is always independent of a constant.
- Independence is an assumption about a model, checkable only within one. "Historically independent" claims are model claims — the correct response is the brittleness-flag: which measure, on which σ-algebra, checked against which data?
Independence is a multiplication property, inherited by everything computed from the parts — and nothing else. If you cannot write the multiplication check, you do not have the independence.
Products built, independence defined and constructible. The next subchapter proves Fubini–Tonelli: the theorem that integrals over product spaces may be computed as iterated integrals — the license for every "integrate out one variable at a time" manipulation in practice, with the counterexamples (order matters, Tonelli's nonnegativity, the summation trap) that show exactly where the license stops.