Fubini and Tonelli: Integrating One Variable at a Time
Every multivariate computation in practice is a stack of single-variable integrals. This subchapter proves when that stacking is legal — and exhibits the double sum where it isn't, which is one line long and famous for a reason.
In this subchapter
6.2.1The claim, and why it needs proof
On a product space (Ω₁ × Ω₂, F₁ ⊗ F₂, P₁ ⊗ P₂) — built in 6.1 — the claim of this subchapter is that
the double integral equals both iterated integrals, in either order. The claim is not obvious for a structural reason: the inner integral ω₁ ↦ ∫ f(ω₁, ·) dP₂ is a function of ω₁ — is it even measurable? Its value is defined by an integral over a fiber (the slice of the product at fixed ω₁), and nothing in Chapter 5's one-variable theory guarantees that a family of fiber-integrals assembles into a measurable function. Worse, the equality itself can fail — §4 — so the proof must produce both measurability and equality, and the hypotheses must be exactly strong enough to bar the failure.
Both theorems run the standard machine one more time — indicators → simple → nonnegative → general — on a product space. The load-bearing rung is now sections: for fixed ω₁, the function fω₁(ω₂) = f(ω₁, ω₂) is a one-variable function on Ω₂, and the machine must track measurability of sections, their integrals, and the assembly. We set up the section formalism in §5 and sketch the ladder; the full ε-management is the same good-sets climb as 5.4 Theorem 1, with rectangles playing the role previously played by level sets.
6.2.2Tonelli: nonnegative stacking is always legal
If f ≥ 0 is F₁⊗F₂-measurable, then: every section is measurable; the section integrals are measurable functions of the fixed variable; and
— no integrability assumption, both orders, values allowed to be +∞. Nonnegativity is the whole hypothesis; nonnegative mass cannot hide, so counting it row-wise, column-wise, or all at once must agree.
Indicators of rectangles: ∫∫ 1A×B dP₂ dP₁ = ∫ 1AP₂(B) dP₁ = P₁(A)P₂(B) = the rectangle formula — the base rung is 6.1 Theorem 1. Simple functions: linearity in both slots. Nonnegative f: banding approximants fn ↑ f; MCT on the fiber (inner integral), MCT on the assembly (outer), MCT for the double integral — all three limits climb together, and monotone limits preserve both the equality and measurability (a monotone limit of measurable functions is measurable, 4.1 Theorem 2). ∎-sketch; the honest gap is verifying the class of functions where all rungs hold is closed under the machine's operations — the good-sets argument with the d-system of "well-behaved f", verbatim the pattern of 5.4.
6.2.3Fubini: integrable stacking is legal
If ∫|f| d(P₁⊗P₂) < ∞ — f integrable on the product — then both iterated integrals exist, are finite, and equal the double integral (and the section integrals exist for a.e. fixed variable, with the exceptional fibers forming a null set). This is the theorem you invoke for signed or complex-valued f; its hypothesis is exactly what Tonelli does not need.
Tonelli on |f| ≥ 0 gives ∫∫|f| dP₂ dP₁ = ∫|f| d(P₁⊗P₂) < ∞, so the function ω₁ ↦ ∫|f(ω₁,·)| dP₂ is finite a.e. — the fibers are integrable for almost every ω₁. Split f = f+ − f−, apply Tonelli to each nonnegative part, subtract: legitimate because both parts have finite integrals (the ∞−∞ warning of 5.1 §5 is disarmed by the hypothesis — this is why the hypothesis is what it is). Linearity in the outer integral assembles the result. ∎
Since checking ∫|f| < ∞ directly on the product is often hard, the standard protocol is: (1) tentatively stack with Tonelli — legal since nonnegative — and (2) if the result is finite, retroactively conclude integrability and license the computation; if it is +∞, conclude nothing about the signed case and re-examine. This "Tonelli-first" discipline resolves 90% of practice: e.g. every E[g(X)h(Y)] = E[g(X)]E[h(Y)] factorization is Tonelli run on the product of laws, and independence of X, Y (6.1) is what reduces the product-measure statement to the two separate one-dimensional integrals — the promise of 6.1 Theorem 2, now delivered in full.
6.2.4The double-sum counterexample
The countable product ℕ × ℕ with counting measure (mass 1 per point) — a σ-finite measure, the natural one. Let
an infinite matrix with 1's on the diagonal, −1's just below it. Row totals: row 1 contains only the diagonal 1 (there is no entry below it), so it sums to 1; every row m ≥ 2 contains the diagonal 1 at n = m and the subdiagonal −1 at n = m−1, so it sums to 0. Hence ΣmΣn amn = 1. Column totals: column n contains 1 at m = n and −1 at m = n+1, so every column sums to 0: ΣnΣm amn = 0. The two iterated sums differ: 1 ≠ 0. Fubini fails, as it must — the hypothesis fails too: Σ|amn| = ∞ (infinitely many ±1's). No contradiction with the theorems: this matrix is exactly the shape excluded by integrability, and its lesson is the one to keep: an iterated computation on a non-integrable object computes the order of summation, not the object.
For a feeling for the analytic version: ∫₀¹∫₀¹ (x²−y²)/(x²+y²)² dx dy — the two iterated integrals evaluate to ±π/4 (an antiderivative computation in closed form); the integrand is not integrable on the square in absolute value, and the sign flips with the order of integration. This is the counterexample behind the classical warning that "improper double integrals" (calculus's workaround) pick up order-dependence that Lebesgue integrability forbids. In a quant's life: double sums over sparse grids of exposures (a.k.a. tail-risk aggregation tables) are exactly where non-absolute summability creeps in — cancellations across a table that the reading order creates.
6.2.5Sections: the honest statement of what Fubini proves
For E ∈ F₁⊗F₂ and fixed ω₁, the section is Eω₁ = { ω₂ : (ω₁, ω₂) ∈ E } ⊆ Ω₂. Sections of measurable sets are measurable — the fact that looks obvious and requires the good-sets argument: the sets whose sections are all measurable form a σ-algebra containing the rectangles, hence contain the product σ-algebra. Everything else in Fubini–Tonelli is built on this rung: indicators of sections, then simple, then MCT. Worth knowing that the "obvious" step is the one that carries the proof's weight — the pattern of this course throughout.
The marginal law of X given a joint law on (X,Y): the joint law ν on ℝ² assigns to a set what the pair assigns; the marginal is μX(A) = ν(A × ℝ) — computed by Tonelli as an integral over a section: integrate the joint density over y for each fixed x. Every "marginalize" operation in statistics is a section computation; Fubini is the theorem that marginalization is well-defined and order-tolerant. And conditional laws (8.1) are the ratios of the same section densities — the machinery is one and the same.
6.2.6Practice: the discipline of iterated computation
(a) A bivariate normal's covariance: E[XY] − E[X]E[Y] computed by stacking — integrable (light tails, 5.5), Fubini licensed, answer ρσ₁σ₂. (b) The pair (X, X²) for symmetric ±1-valued X: E[X·X²] = E[X³] = 0 = E[X]E[X²] = 0 — uncorrelated, and maximally dependent (each determines the other). Uncorrelated pairs pass one Fubini computation; independence (6.1) requires every computation to factor. The example earns its place as the sharpest one-line separation of the two concepts in the course.
- Nonnegative integrand? Stack freely (Tonelli), but a +∞ verdict licenses nothing further.
- Signed? You must establish ∫|f| < ∞ — or work with the Tonelli-first protocol honestly (stack |f|, confirm finite).
- Order of integration is a computational choice only under a theorem's license. Otherwise it is a mathematical act with consequences (Example 1: the answer 1 vs 0 was created by the order).
- σ-finiteness of the measures (counting measure on ℕ is fine; a non-σ-finite measure breaks Tonelli too — a subtler failure we flag but do not exhibit) is a standing hypothesis of both theorems; it holds for every law in practice.
Stack your integrals in any order — if the mass is nonnegative or absolutely finite. If neither, the order is not bookkeeping; it is the answer's author.
The product machinery is complete: pairs of variables, joint laws, iterated computation. The next subchapter turns to the infinite run: what happens to a sequence of events as n → ∞ — the Borel–Cantelli lemmas, proved with the tools of 2.2 and 6.1, with the counterexample that shows independence is genuinely needed for the second lemma, and the meaning of "infinitely often" made precise via limsup of events.