Fubini-Tonelli Theorem and Product Measures

Intuition Beginner

A double integral computes the volume under a surface drawn over a flat rectangle. There are two natural ways to compute that volume. The first slices the rectangle into thin strips parallel to one axis, computes the area under each strip, and adds the strip areas. The second slices the rectangle into thin strips parallel to the other axis and adds those. Both procedures should give the same total volume, because the volume of a solid does not depend on the direction you cut it from. This is the picture behind the Fubini-Tonelli theorem.

In symbols, the theorem says you can compute a double integral as an iterated single integral, in either order. If you integrate over the $x$ variable first while $y$ is held fixed, you get a function of $y$ alone; integrating that function over $y$ gives the same answer as integrating $y$ first and then $x$. The two iterated integrals match the single double integral, and they match each other. This is one of the most useful structural facts in integration theory, because it turns a problem in two variables into a sequence of two problems in one variable.

There are two distinct versions of the theorem. Tonelli's version handles functions that take non-negative values: it says the equality of the iterated integrals and the double integral always holds, with no extra condition needed beyond the function being measurable. Fubini's version handles functions that can take positive and negative values, or complex values: it says the equality holds when the function is absolutely integrable, that is when the integral of the absolute value is finite. The combined Fubini-Tonelli strategy is to first check absolute integrability using Tonelli on the absolute value, then apply Fubini to get the iterated-integral identity for the original signed function.

The theorem has one important fine-print clause: both measures involved must be sigma-finite. A measure is sigma-finite if the whole space can be split into countably many pieces each of finite measure. Lebesgue measure on the real line is sigma-finite, because the real line decomposes into the intervals from minus $n$ to plus $n$ for integers $n$. Counting measure on an uncountable set is not sigma-finite, and counterexamples to Fubini exist in that setting.

The one-sentence takeaway: a double integral equals either iterated integral, provided the function is non-negative (Tonelli) or absolutely integrable (Fubini) and the two underlying measures are sigma-finite.

Visual Beginner

Imagine a loaf of bread sitting on a cutting board. The board is the rectangle in the plane, and the bread above the board represents the function whose double integral we want. There are two ways to compute the volume of the bread.

Left: slice the loaf into thin wafers parallel to the $y$-axis, weigh each wafer (compute its area), and add the wafer weights. Right: slice the loaf into thin wafers parallel to the $x$-axis, weigh each, and add. Both procedures give the same volume.

The mathematical statement that the two slicing strategies match is the iterated-integral identity in Fubini-Tonelli. The non-negative version (Tonelli) is the bread picture without restrictions. The general version (Fubini) requires the loaf to have finite total weight, that is the absolute integral to be finite, so the slicing procedure does not run into infinite cancellations.

Worked example Beginner

We compute a double integral two ways and confirm the answers match.

Step 1. Let $f(x,y)=xy$ on the unit square $[0,1]\times [0,1]$. The function is continuous and non-negative on the square, so Tonelli's theorem applies.

Step 2. Integrate in $x$ first. Fix $y$ and integrate $xy$ over $x$ from $0$ to $1$. Since $y$ is a constant during the inner integration, the inner integral is $y$ times the integral of $x$ from $0$ to $1$, which is $y\cdot (1/2)=y/2$. Now integrate the resulting function $y/2$ over $y$ from $0$ to $1$. The answer is $(1/2)\cdot (1/2)=1/4$.

Step 3. Integrate in $y$ first. Fix $x$ and integrate $xy$ over $y$ from $0$ to $1$. The inner integral is $x$ times the integral of $y$ from $0$ to $1$, which is $x\cdot (1/2)=x/2$. Integrate $x/2$ over $x$ from $0$ to $1$ to get $(1/2)\cdot (1/2)=1/4$.

Step 4. Compare. Both iterated integrals give $1/4$. The double integral of $xy$ over the unit square is $1/4$. The two slicing orders agree, as Tonelli predicts.

Step 5. A signed example. Let $g(x,y)=x-y$ on the unit square. The function changes sign across the diagonal $y=x$. The absolute value $|x-y|$ is bounded by $1$ on the square, so $|g|$ has integral at most $1$ and Fubini applies. Integrating $g$ in either order: integrating $x-y$ over $x$ from $0$ to $1$ gives $(1/2)-y$; integrating that over $y$ gives $(1/2)-(1/2)=0$. The other order gives the same answer by symmetry. The double integral vanishes, matching the visual antisymmetry of the function $x-y$ under the swap $(x,y)\mapsto (y,x)$.

What this tells us: when the function is non-negative or absolutely integrable, the iterated integrals give the same answer in either order, and that common answer equals the double integral. The procedure of computing one variable at a time is the standard tool for evaluating multiple integrals in calculus and probability.

Check your understanding Beginner

Formal definition Intermediate+

Let $(X,\mathcal{M},\mu )$ and $(Y,\mathcal{N},\nu )$ be measure spaces.

Definition (product $\sigma$-algebra). The product $\sigma$-algebra $\mathcal{M}\otimes \mathcal{N}$ on the Cartesian product $X\times Y$ is the smallest $\sigma$-algebra containing every measurable rectangle $A\times B$ with $A\in \mathcal{M}$, $B\in \mathcal{N}$. The class of measurable rectangles is a semiring, and the product $\sigma$-algebra is its $\sigma$-algebra generated.

Definition (product measure). If $\mu$ and $\nu$ are $\sigma$-finite, there is a unique measure $\mu \otimes \nu$ on $(X\times Y,\mathcal{M}\otimes \mathcal{N})$ satisfying $(\mu \otimes \nu )(A\times B)=\mu (A)\cdot \nu (B)$ for every measurable rectangle. (We use the convention $0\cdot \infty =0$.) Existence is proved via Carathéodory's extension theorem applied to the additive set-function $\mu \otimes \nu$ on the algebra of finite disjoint unions of measurable rectangles; uniqueness is from $\sigma$-finiteness via the Hahn-Kolmogorov uniqueness theorem.

Definition (sections). For $E\subseteq X\times Y$ and $x\in X$, the $x$-section of $E$ is $E_x=\{y\in Y:(x,y)\in E\}$. For $y\in Y$, the $y$-section is $E^y=\{x\in X:(x,y)\in E\}$. For a function $f:X\times Y\to \mathbb{C}$, the $x$-section $f_x:Y\to \mathbb{C}$ is the map $y\mapsto f(x,y)$, and the $y$-section $f^y$ is $x\mapsto f(x,y)$.

Definition (iterated integral). For a non-negative measurable function $f:X\times Y\to [0,\infty ]$, the iterated integral in the order $(y,x)$ is $${\int }_X\left({\int }_{Y}f(x,y)d\nu (y)\right)d\mu (x).$$ The integrand is measurable in $x$ once $\nu$ is $\sigma$-finite (a lemma in the construction of the product measure). The reverse order is defined symmetrically.

Counterexamples to common slips Intermediate+

• Iterated integrals can differ when $\sigma$-finiteness fails. On $X=Y=[0,1]$ with $\mu$ Lebesgue and $\nu$ counting measure, the indicator $f={\chi }_{\Delta }$ of the diagonal $\Delta =\{(x,x):x\in [0,1]\}$ has ${\int }_X{\int }_{Y}fd\nu d\mu =0$ (each $x$-section is the single point $\{x\}$, of $\mu$-measure zero pulled outside) and ${\int }_Y{\int }_{X}fd\mu d\nu ={\int }_{Y}0d\nu =0$ — wait, both are zero here, but the actual Sierpiński-style counterexample uses a non-measurable diagonal-type set and requires the continuum hypothesis. The cleaner failure: on $[0,1]\times [0,1]$ with $\mu$ Lebesgue and $\nu$ counting, the function $f(x,y)={\chi }_{\{x=y\}}$ has one iteration giving $0$ and the other giving $1$, because the inner $\nu$-integral over $\{y=x\}$ is $1$ while the inner $\mu$-integral over a single $\{x=y\}$ slice is $0$.

• Tonelli does not require integrability. For non-negative measurable $f$, Tonelli says ${\int }_{X\times Y}fd(\mu \otimes \nu )={\int }_X{\int }_{Y}fd\nu d\mu$ as elements of $[0,\infty ]$, with both sides possibly $+\infty$ simultaneously. Fubini's hypothesis (absolute integrability) is needed only when $f$ is signed or complex-valued.

• Absolute integrability on the product is the right hypothesis for signed Fubini. The condition ${\int }_X{\int }_Y|f|d\nu d\mu <\infty$ (or equivalently ${\int }_{X\times Y}|f|d(\mu \otimes \nu )<\infty$, by Tonelli on $|f|$) suffices. Mere existence of the iterated integrals of $f$ itself is not enough: examples like $f(x,y)=(x^2-y^2)/(x^2+y^2{)}^2$ on $(0,1{)}^2$ give different iterated integrals because $|f|$ is not integrable.

• Sections of a measurable set need not all be measurable. The product $\sigma$-algebra $\mathcal{M}\otimes \mathcal{N}$ may be strictly smaller than the Lebesgue-completion of the product measure. If $E\in \mathcal{M}\otimes \mathcal{N}$, every section is measurable (Theorem 1 below). But if $E$ is in the completion of $\mu \otimes \nu$ but not in $\mathcal{M}\otimes \mathcal{N}$, some sections of $E$ may fail to be measurable — though the set of bad $x$'s has $\mu$-measure zero. Folland §2.5 discusses this completion subtlety.

• The product measure on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ is not always equal to Lebesgue measure on ${\mathbb{R}}^{m+n}$. The Borel product ${\mathcal{B}}_m\otimes {\mathcal{B}}_n$ equals ${\mathcal{B}}_{m+n}$, and the Borel product measure equals Lebesgue measure restricted to Borel sets. But the Lebesgue completion in dimension $m+n$ strictly contains the Lebesgue product of the lower-dimensional completions; the difference is irrelevant for integration but matters for set-theoretic statements.

Key theorem with proof Intermediate+

Theorem (Tonelli; Tonelli 1909 Rend. Acc. Lincei 18, 246). Let $(X,\mathcal{M},\mu )$ and $(Y,\mathcal{N},\nu )$ be $\sigma$-finite measure spaces. For every non-negative $\mathcal{M}\otimes \mathcal{N}$-measurable function $f:X\times Y\to [0,\infty ]$:

(a) For every $x\in X$ the section $f_x$ is $\mathcal{N}$-measurable; for every $y\in Y$ the section $f^y$ is $\mathcal{M}$-measurable.

(b) The function $x\mapsto {\int }_Y{f}_{x}d\nu$ is $\mathcal{M}$-measurable; the function $y\mapsto {\int }_X{f}^{y}d\mu$ is $\mathcal{N}$-measurable.

(c) The product integral equals either iterated integral: $${\int }_{X\times Y}fd(\mu \otimes \nu )={\int }_X\left({\int }_{Y}f(x,y)d\nu (y)\right)d\mu (x)={\int }_Y\left({\int }_{X}f(x,y)d\mu (x)\right)d\nu (y).$$

Proof. We use the standard machine: indicator functions $\to$ non-negative simple functions $\to$ non-negative measurable functions, with the monotone convergence theorem 02.07.04 performing the limit step.

Step 1 (indicators of measurable rectangles). Let $f={\chi }_{A\times B}$ for $A\in \mathcal{M}$, $B\in \mathcal{N}$. Then $f_x={\chi }_A(x){\chi }_B$ (the indicator of $B$ multiplied by the constant ${\chi }_A(x)\in \{0,1\}$), which is $\mathcal{N}$-measurable. The inner integral is ${\int }_Y{f}_{x}d\nu ={\chi }_A(x)\cdot \nu (B)$, which is $\mathcal{M}$-measurable in $x$. The outer integral is ${\int }_X{\chi }_A(x)\nu (B)d\mu =\mu (A)\cdot \nu (B)=(\mu \otimes \nu )(A\times B)={\int }_{X\times Y}fd(\mu \otimes \nu )$. Similarly for the reverse iteration. The identity holds on indicators of rectangles.

Step 2 (indicators of measurable sets — the monotone class step). Let $\mathcal{C}$ be the collection of sets $E\in \mathcal{M}\otimes \mathcal{N}$ for which (a), (b), (c) hold for $f={\chi }_E$. By Step 1, $\mathcal{C}$ contains every measurable rectangle. We show $\mathcal{C}$ is a monotone class.

Sub-step 2a (finite disjoint unions). If $E_1,E_2\in \mathcal{C}$ are disjoint with both in $\mathcal{C}$, then ${\chi }_{E_1\cup E_2}={\chi }_{E_1}+{\chi }_{E_2}$, sections distribute over the sum, and integrals of non-negative functions are additive. So $E_1\cup E_2\in \mathcal{C}$ — and by induction the algebra of finite disjoint unions of rectangles is in $\mathcal{C}$.

Sub-step 2b (monotone limits). If $E_n\in \mathcal{C}$ with $E_n\uparrow E$, then ${\chi }_{E_n}\uparrow {\chi }_E$. Each $({\chi }_{E_n}{)}_x\uparrow ({\chi }_E{)}_x$ pointwise in $y$, so $({\chi }_E{)}_x$ is $\mathcal{N}$-measurable as a pointwise increasing limit of measurable functions. By the monotone convergence theorem ${\int }_Y({\chi }_{E_n}{)}_{x}d\nu \uparrow {\int }_Y({\chi }_E{)}_{x}d\nu$ in $x$, so the limit function is $\mathcal{M}$-measurable. Applying MCT again in the outer variable, ${\int }_X{\int }_Y({\chi }_E{)}_{x}d\nu d\mu ={\lim }_n{\int }_X{\int }_Y({\chi }_{E_n}{)}_{x}d\nu d\mu ={\lim }_n(\mu \otimes \nu )(E_n)=(\mu \otimes \nu )(E)$ (the last equality from continuity from below of $\mu \otimes \nu$).

Sub-step 2c (monotone decreasing limits, assuming $\sigma$-finiteness). The corresponding statement for $E_n\downarrow E$ requires the finiteness condition $(\mu \otimes \nu )(E_1)<\infty$ to interchange limit and integral via the dominated convergence theorem 02.07.05 with dominator ${\chi }_{E_1}$. By $\sigma$-finiteness, $X\times Y$ is a countable union of measurable rectangles $X_k\times Y_k$ with $\mu (X_k),\nu (Y_k)<\infty$; restricting to each piece, the monotone-decreasing case is handled by DCT, and the global statement is recovered by taking $k\to \infty$.

Sub-step 2d (monotone class theorem). The monotone class theorem (Halmos 1950 §6, Theorem B; Bogachev 2007 Volume 1, Theorem 2.10.4) says: a monotone class containing an algebra contains the $\sigma$-algebra generated by that algebra. The algebra of finite disjoint unions of measurable rectangles generates $\mathcal{M}\otimes \mathcal{N}$, and we have shown $\mathcal{C}$ is a monotone class containing this algebra (sub-steps 2a, 2b, 2c). Hence $\mathcal{C}=\mathcal{M}\otimes \mathcal{N}$, and the identity holds for every indicator function of a measurable set.

Step 3 (non-negative simple functions). A non-negative simple function $s$ is a finite non-negative linear combination $s={\sum }_{k=1}^N{c}_k{\chi }_{E_k}$ with $E_k\in \mathcal{M}\otimes \mathcal{N}$. Sections of $s$ are non-negative linear combinations of sections of the ${\chi }_{E_k}$, hence measurable. The iterated and product integrals are linear in the non-negative coefficients $c_k$, and the identity for each ${\chi }_{E_k}$ (Step 2) gives the identity for $s$.

Step 4 (non-negative measurable functions — the MCT step). Let $f:X\times Y\to [0,\infty ]$ be measurable. By the standard simple-function approximation theorem 02.07.03, there is an increasing sequence of non-negative simple functions $s_n\uparrow f$ pointwise on $X\times Y$. For each $x$, $(s_n{)}_x\uparrow f_x$ pointwise, so $f_x$ is $\mathcal{N}$-measurable as a pointwise increasing limit. By MCT in the inner variable, ${\int }_Y(s_n{)}_{x}d\nu \uparrow {\int }_Y{f}_{x}d\nu$, so the inner-integral function (in $x$) is the pointwise limit of $\mathcal{M}$-measurable functions and hence $\mathcal{M}$-measurable. By MCT in the outer variable, $${\int }_X{\int }_Y{f}_{x}d\nu d\mu =\underset{n}{\lim }{\int }_X{\int }_Y(s_n{)}_{x}d\nu d\mu =\underset{n}{\lim }{\int }_{X\times Y}{s}_{n}d(\mu \otimes \nu )={\int }_{X\times Y}fd(\mu \otimes \nu ).$$ The reverse iteration follows by a symmetric argument. The three quantities agree.

$\square$

Bridge. Tonelli's theorem builds toward Fubini (Theorem 5 below) by providing the integrability test on $|f|$: a signed or complex measurable function $f$ on $X\times Y$ is absolutely integrable on the product if and only if either iterated integral of $|f|$ is finite, and this equivalence is exactly the Tonelli identity applied to $|f|$. The combined Fubini-Tonelli strategy is therefore to use Tonelli on $|f|$ to confirm finite total absolute mass, then use Fubini on $f$ to extract the iterated-integral identity. The foundational reason the two-step strategy works is the standard machine of measure theory: identities that hold on indicators of rectangles extend to indicators of $\sigma$-algebra-generated sets via the monotone class theorem, then to non-negative simple functions by non-negative linearity, then to non-negative measurable functions by MCT, and finally to signed/complex functions by linearity over the positive-and-negative-part decomposition $f=f^{+}-f^{-}$ (or the four-piece decomposition for complex $f$). The bridge is between the rectangle formula $(\mu \otimes \nu )(A\times B)=\mu (A)\nu (B)$ and the iterated-integral identity for arbitrary measurable functions, with the monotone class theorem playing the load-bearing role in each step. The pattern recurs in Bochner integration on product spaces, in distribution-theoretic Schwartz kernel theorems, and in probabilistic constructions of independent product processes.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: partial — Mathlib provides the central Fubini-Tonelli architecture. MeasureTheory.Measure.prod constructs the product measure $\mu \otimes \nu$ for $\sigma$-finite $\mu ,\nu$. MeasureTheory.lintegral_prod is Tonelli on the lower-Lebesgue integral. MeasureTheory.integral_prod is Fubini on Bochner-integrable functions. MeasureTheory.MeasurePreserving and MeasureTheory.Measure.prod_apply give the rectangle formula. The Sierpiński-style counterexamples in the non-$\sigma$-finite setting are not formalised in Mathlib but are recorded in the unit's Master tier and the companion module Codex.Analysis.MeasureTheory.FubiniTonelli.

import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.Prod

open MeasureTheory

variable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
variable (μ : Measure α) (ν : Measure β)
variable [SigmaFinite μ] [SigmaFinite ν]

-- Tonelli for lower-Lebesgue integrals on σ-finite product measures
example (f : α × β → ENNReal) (hf : Measurable f) :
    ∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ :=
  lintegral_prod f hf.aemeasurable

-- Fubini for Bochner-integrable functions
example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
    ∫ z, f z ∂(μ.prod ν) = ∫ x, ∫ y, f (x, y) ∂ν ∂μ :=
  integral_prod f hf

Advanced results Master

The advanced theory of product measures and the Fubini-Tonelli identity splits across seven strands: the product $\sigma$-algebra and rectangle formula, the monotone-class construction of the product measure, Tonelli for non-negative measurable functions, Fubini for $L^1$-functions, completion subtleties, counterexamples without $\sigma$-finiteness, and applications to convolution, Cavalieri, and integral-transform identities.

Theorem 1 (Measurability of sections; Folland 2e Proposition 2.34). Let $(X,\mathcal{M})$ and $(Y,\mathcal{N})$ be measurable spaces, and let $E\in \mathcal{M}\otimes \mathcal{N}$. For every $x\in X$, the section $E_x\in \mathcal{N}$; for every $y\in Y$, the section $E^y\in \mathcal{M}$. For a $\mathcal{M}\otimes \mathcal{N}$-measurable function $f:X\times Y\to \overline{\mathbb{R}}$, the section $f_x$ is $\mathcal{N}$-measurable for every $x$ and $f^y$ is $\mathcal{M}$-measurable for every $y$.

The section measurability is the load-bearing structural fact behind Fubini-Tonelli: the iterated integrals are defined only because the sectioned functions are measurable. The proof is the standard machine: the class $\mathcal{C}=\{E\in \mathcal{M}\otimes \mathcal{N}:E_x\in \mathcal{N}\text{ for all }x\}$ contains every measurable rectangle (since $(A\times B{)}_x=B$ if $x\in A$, $=\varnothing$ otherwise), is closed under complements and countable unions (by the corresponding section operations), hence is a $\sigma$-algebra containing the rectangles. So $\mathcal{C}=\mathcal{M}\otimes \mathcal{N}$. The function statement follows from the level-set characterisation $\{f_x>c\}=(\{f>c\}{)}_x$ ^{[Folland Real Analysis 2e]}.

Theorem 2 (Existence and uniqueness of the product measure; Halmos 1950 §35; Bogachev 2007 Vol. 1 Theorem 3.3.1). Let $(X,\mathcal{M},\mu )$ and $(Y,\mathcal{N},\nu )$ be $\sigma$-finite measure spaces. There exists a unique measure $\mu \otimes \nu$ on $(X\times Y,\mathcal{M}\otimes \mathcal{N})$ with $(\mu \otimes \nu )(A\times B)=\mu (A)\cdot \nu (B)$ for all $A\in \mathcal{M}$, $B\in \mathcal{N}$ (with $0\cdot \infty =0$).

The existence proof uses Carathéodory's extension theorem. The set-function $\mu \otimes \nu$ defined on the semiring of measurable rectangles by $(\mu \otimes \nu )(A\times B)=\mu (A)\nu (B)$ extends additively to the algebra of finite disjoint unions of measurable rectangles. The countable-additivity of this extension on the algebra is proved by: given a countable disjoint cover $E={⨆}_n{A}_n\times B_n$, the sectional identity ${\chi }_E(x,y)={\sum }_n{\chi }_{A_n}(x){\chi }_{B_n}(y)$ combined with monotone convergence in the inner $y$-integral gives the formula $(\mu \otimes \nu )(E)={\sum }_n\mu (A_n)\nu (B_n)$. Carathéodory's outer-measure-to-measure construction lifts the algebra-additive set-function to a measure on the generated $\sigma$-algebra $\mathcal{M}\otimes \mathcal{N}$.

The uniqueness uses the Hahn-Kolmogorov uniqueness theorem (also called the $\pi$-$\lambda$ theorem or the monotone class theorem): a $\sigma$-finite measure on a $\sigma$-algebra is determined by its values on a generating $\pi$-system (a class closed under finite intersections). The measurable rectangles form a $\pi$-system generating $\mathcal{M}\otimes \mathcal{N}$, so any other measure agreeing with $\mu \otimes \nu$ on rectangles is identical ^{[Halmos 1950]}.

Theorem 3 (Tonelli; Tonelli 1909 Rend. Acc. Lincei 18, 246). Let $\mu ,\nu$ be $\sigma$-finite. For any non-negative $\mathcal{M}\otimes \mathcal{N}$-measurable function $f:X\times Y\to [0,\infty ]$, $${\int }_{X\times Y}fd(\mu \otimes \nu )={\int }_X\left({\int }_{Y}f(x,y)d\nu (y)\right)d\mu (x)={\int }_Y\left({\int }_{X}f(x,y)d\mu (x)\right)d\nu (y),$$ with all three expressions belonging to $[0,\infty ]$.

Tonelli's 1909 Rendiconti dell'Accademia dei Lincei paper ^{[Tonelli 1909]} introduced the non-negative-function version of the iterated-integral identity, building on Fubini's 1907 version for absolutely integrable functions. Tonelli's contribution was the recognition that absolute integrability was not needed when the function is non-negative — the equality of the three quantities holds unconditionally for non-negative measurable functions on $\sigma$-finite product spaces, possibly with all three being $+\infty$. The proof is the standard machine (Step 1-4 of the proof of Theorem above).

Theorem 4 (Fubini; Fubini 1907 Rend. Acc. Lincei 16, 608). Let $\mu ,\nu$ be $\sigma$-finite, and $f:X\times Y\to \mathbb{C}$ (or $\mathbb{R}$ or $\overline{\mathbb{R}}$) be $\mu \otimes \nu$-integrable. Then:

(a) For $\mu$-a.e. $x\in X$, the section $f_x\in L^1(\nu )$; for $\nu$-a.e. $y\in Y$, the section $f^y\in L^1(\mu )$.

(b) The functions $x\mapsto {\int }_Y{f}_{x}d\nu$ (defined $\mu$-a.e.) and $y\mapsto {\int }_X{f}^{y}d\mu$ (defined $\nu$-a.e.) belong to $L^1(\mu )$ and $L^1(\nu )$ respectively.

(c) The product and iterated integrals coincide: $${\int }_{X\times Y}fd(\mu \otimes \nu )={\int }_X{\int }_{Y}f(x,y)d\nu (y)d\mu (x)={\int }_Y{\int }_{X}f(x,y)d\mu (x)d\nu (y).$$

Fubini's 1907 paper ^{[Fubini 1907]} proved the iterated-integral identity for absolutely integrable real-valued functions; the result was the first general statement of the iterated-integral interchange beyond the Cauchy-Riemann context, building on earlier special cases (Cauchy 1814 for continuous functions on rectangles, Lebesgue 1902 for bounded measurable functions). The proof reduces to Tonelli on $f^{+},f^{-}$ (positive and negative parts), using $|f|=f^{+}+f^{-}\in L^1$ to control both parts and then combining via linearity of the integral. The complex case decomposes into real and imaginary parts and applies the real case to each.

Theorem 5 (Fubini-Tonelli combined; Folland 2e Theorem 2.37). Let $\mu ,\nu$ be $\sigma$-finite, $f:X\times Y\to \mathbb{C}$ be measurable. If either of the iterated integrals $${\int }_X{\int }_Y|f(x,y)|d\nu (y)d\mu (x),{\int }_Y{\int }_X|f(x,y)|d\mu (x)d\nu (y)$$ is finite, then $f\in L^1(\mu \otimes \nu )$ and Fubini applies.

The combined statement codifies the standard practical strategy: apply Tonelli to $|f|$ to check absolute integrability via one iteration order; if finite, conclude $f\in L^1(\mu \otimes \nu )$ and apply Fubini to $f$ to get the iterated-integral identity for $f$ itself. This is the procedure invoked in virtually every real-analytic computation involving multiple integrals or signed integrands.

Theorem 6 (Completion and a.e. measurability; Folland 2e Theorem 2.39). Let $\overline{\mu \otimes \nu }$ denote the completion of $\mu \otimes \nu$, that is the measure obtained by Carathéodory completion on the $\sigma$-algebra of $(\mu \otimes \nu )$-measurable sets in the sense of Lebesgue. Then:

(a) If $f$ is $\overline{\mu \otimes \nu }$-measurable, the section $f_x$ is $\overline{\nu }$-measurable for $\mu$-a.e. $x$ (and analogously for $f^y$). The set of bad $x$'s is contained in a $\mu$-null set.

(b) Tonelli and Fubini hold with $\overline{\mu \otimes \nu }$ in place of $\mu \otimes \nu$, with the inner integrals understood as integrals of $\overline{\nu }$-measurable functions on the full-$\mu$-measure set of good $x$'s, and the integrand defined arbitrarily (e.g., to be zero) on the $\mu$-null exceptional set.

The completion subtlety matters when working with Lebesgue measure on ${\mathbb{R}}^{m+n}$ rather than with the Borel product measure: ${\mathcal{B}}_{m+n}={\mathcal{B}}_m\otimes {\mathcal{B}}_n$, but the Lebesgue $\sigma$-algebra ${\mathcal{L}}_{m+n}$ is strictly larger than ${\mathcal{L}}_m\otimes {\mathcal{L}}_n$. The Fubini-Tonelli statements for Lebesgue-measurable integrands rely on Theorem 6, with "for $\mu$-a.e. $x$" appearing in the conclusions about section measurability.

Theorem 7 (Sierpiński counterexample without $\sigma$-finiteness; Sierpiński 1920 Fundamenta Math. 1, 112). Let $X=Y=[0,1]$ with $\mu$ Lebesgue measure and $\nu$ counting measure (which is not $\sigma$-finite on the uncountable space $[0,1]$). Let $f={\chi }_{\Delta }$ where $\Delta =\{(x,x):x\in [0,1]\}$ is the diagonal. Then:

(a) For every $x$, the section $f_x$ on $Y$ is the indicator of the single point $\{x\}$, with counting-measure integral ${\int }_Y{f}_{x}d\nu =1$. So ${\int }_X{\int }_Y{f}_{x}d\nu d\mu ={\int }_{X}1dm=1$.

(b) For every $y$, the section $f^y$ on $X$ is the indicator of $\{y\}$, with Lebesgue-measure integral ${\int }_X{f}^{y}dm=0$. So ${\int }_Y{\int }_X{f}^{y}dmd\nu ={\int }_{Y}0d\nu =0$.

The two iterated integrals are $1$ and $0$, unequal. The Fubini-Tonelli identity fails because $\nu$ is not $\sigma$-finite. Sierpiński's 1920 Fundamenta Mathematicae paper ^{[Sierpiński 1920]} gave a more elaborate continuum-hypothesis-dependent counterexample in which both iterated integrals exist as Lebesgue integrals but give different values; the present simpler example uses non-$\sigma$-finite counting measure to exhibit the same phenomenon without set-theoretic axioms beyond ZFC.

Theorem 8 (Convolution and Young's inequality; Young 1912 Proc. London Math. Soc. (2) 11, 357). For $f\in L^p({\mathbb{R}}^n)$ and $g\in L^q({\mathbb{R}}^n)$ with $1\le p,q,r\le \infty$ and $1/p+1/q=1+1/r$, the convolution $$(f\ast g)(x)={\int }_{{\mathbb{R}}^n}f(x-y)g(y)dy$$ is well-defined for a.e. $x$ and satisfies the Young inequality $\Vert f\ast g{\Vert }_r\le \Vert f{\Vert }_p\Vert g{\Vert }_q$.

The proof in the $(1,1,1)$ case uses Tonelli on $|f(x-y)||g(y)|$ and translation-invariance of Lebesgue measure to deduce $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$, the foundational fact that $L^1({\mathbb{R}}^n)$ is a Banach algebra under convolution. The $(p,q,r)$ case follows by Riesz-Thorin interpolation between the endpoints $(1,1,1)$, $(1,p,p)$, and $(p,p^{\prime },\infty )$ for the conjugate exponent $p^{\prime }=p/(p-1)$. The sharp constant in Young's inequality on ${\mathbb{R}}^n$ is due to Brascamp-Lieb 1976 Adv. Math. 20, 151, with Gaussian extremisers ^{[Young 1912]}.

Theorem 9 (Cavalieri principle / layer-cake formula; Cavalieri 1635, Bogachev 2007 Vol. 1 Proposition 2.5.7). For a non-negative measurable function $f$ on $(X,\mathcal{M},\mu )$ and any $p>0$, $${\int }_X{f}^{p}d\mu =p{\int }_0^{\infty }{t}^{p-1}\mu (\{f>t\})dt.$$

The Cavalieri principle (Cavalieri 1635 Geometria indivisibilibus ^{[Cavalieri 1635]}) was the seventeenth-century forerunner of integration: the area under a curve equals the integral of the horizontal-slice widths. In modern form, the layer-cake formula is a one-line Tonelli computation: $f^p(x)=p{\int }_0^{f(x)}{t}^{p-1}dt=p{\int }_0^{\infty }{t}^{p-1}{\chi }_{\{f(x)>t\}}dt$, and Tonelli on the non-negative product $p\cdot t^{p-1}{\chi }_{\{(x,t):f(x)>t\}}$ exchanges the order of integration to give ${\int }_X{f}^{p}d\mu =p{\int }_0^{\infty }{t}^{p-1}\mu (\{f>t\})dt$. The $p=1$ case is the original Cavalieri statement; the general $p$ produces the distributional formula for $L^p$-norms in terms of measure of super-level sets, the cornerstone of weak-$L^p$ theory and the Lorentz spaces $L^{p,q}$ (Lorentz 1950).

Theorem 10 (Volume of $n$-ball; Bogachev 2007 Vol. 1 Example 3.4.7). The Lebesgue measure of the unit ball $B^n=\{x\in {\mathbb{R}}^n:|x|\le 1\}$ is $$\mathrm{vol}(B^n)=\frac{{\pi }^{n/2}}{\Gamma (n/2+1)}.$$

The formula is obtained by Tonelli plus the Gaussian-integral identity (Exercise 8). Writing ${\int }_{{\mathbb{R}}^n}{e}^{-|x{|}^2}dx={\pi }^{n/2}$ via the product of $n$ one-dimensional Gaussians, then re-expressing the integral via the layer-cake formula on $f(x)=e^{-|x{|}^2}$ with $t=e^{-r^2}$, gives ${\pi }^{n/2}={\int }_0^{\infty }n{\omega }_{n-1}{r}^{n-1}{e}^{-r^2}dr/n$ — where ${\omega }_{n-1}=\mathrm{vol}(S^{n-1})$ is the surface area of the unit sphere — leading via the substitution $u=r^2$ to ${\omega }_{n-1}\Gamma (n/2)/2={\pi }^{n/2}$, hence ${\omega }_{n-1}=2{\pi }^{n/2}/\Gamma (n/2)$ and $\mathrm{vol}(B^n)={\omega }_{n-1}/n={\pi }^{n/2}/\Gamma (n/2+1)$.

The recursion $\mathrm{vol}(B^n)=\mathrm{vol}(B^{n-1})\cdot {\int }_{-1}^1(1-x^2{)}^{(n-1)/2}dx$, derived directly from Tonelli by slicing the $n$-ball along the last coordinate, gives an alternative proof and the Wallis-product connection.

Theorem 11 (Plancherel and Fourier inversion via Fubini; Plancherel 1910 Rend. Circ. Mat. Palermo 30, 289). For $f\in L^1\cap L^2({\mathbb{R}}^n)$, the Fourier transform $\hat{f}(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix\cdot \xi }dx$ satisfies $\Vert \hat{f}{\Vert }_2=\Vert f{\Vert }_2$ (Plancherel identity), extending by density to a unitary operator $\mathcal{F}:L^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)$.

The Plancherel identity proof relies on Fubini-Tonelli applied to the function $f(x)g(y)e^{-2\pi i(x-y)\cdot \xi }$ on ${\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n$ (in the variables $x,y,\xi$). Two of the three Fubini interchanges produce the identities $\int f\bar{g}=\int \hat{f}\bar{\hat{g}}$ (the $L^2$-inner-product preservation) and the Fourier-inversion formula $f(x)=\int \hat{f}(\xi )e^{2\pi ix\cdot \xi }d\xi$ for Schwartz $f$. The boundedness of Fubini's interchange is justified by the absolute integrability of the triple-product on ${\mathbb{R}}^{3n}$ under the Schwartz-class hypothesis ^{[Plancherel 1910 not declared; see Hausdorff 1923 placeholder]}.

Synthesis. The Fubini-Tonelli architecture is the foundational reason that modern integration theory can reduce multi-dimensional integrals to iterated one-dimensional integrals, and conversely that iterated-integral identities can be lifted to genuine multi-dimensional statements. The central insight is the rectangle formula $(\mu \otimes \nu )(A\times B)=\mu (A)\nu (B)$ on the semiring of measurable rectangles, extending via Carathéodory and the monotone class theorem to a $\sigma$-additive measure $\mu \otimes \nu$ on the product $\sigma$-algebra, and the resulting iterated-integral identity holding for non-negative measurable functions (Tonelli) and absolutely integrable functions (Fubini) on $\sigma$-finite product spaces.

The pattern generalises in four directions. First, to convolutions: Tonelli is the load-bearing step in proving that $L^1\times L^1\to L^1$ convolution is bounded, identifying $L^1({\mathbb{R}}^n)$ as a Banach algebra. Young's interpolated inequality $L^p\times L^q\to L^r$ for $1/p+1/q=1+1/r$ follows by Riesz-Thorin from this endpoint. Second, to Cavalieri / layer-cake formulas: Tonelli on ${\chi }_{\{(x,t):f(x)>t\}}$ converts integrals into measure-of-super-level-set integrals, the foundation of weak-$L^p$ theory and Lorentz spaces. Third, to Fourier analysis: the Plancherel identity, Fourier inversion, and the convolution-product theorem $\widehat{f\ast g}=\hat{f}\cdot \hat{g}$ all rely on Fubini interchanges in the convolution-Fourier triple-integral. Fourth, to probability theory: independent random variables are precisely those whose joint distribution is a product measure, and Fubini-Tonelli is the foundational fact behind $\mathbb{E}[fg]=\mathbb{E}[f]\mathbb{E}[g]$ for independent integrable $f,g$, the expectation-factorisation identity underlying the strong law of large numbers, the central limit theorem, and Wiener's chaos decomposition.

Putting these together identifies Fubini-Tonelli as the bridge between Carathéodory's outer-measure construction, the standard-machine proof template (indicators-to-simple-to-non-negative-to-signed via MCT and linearity), the convolution algebra structure of $L^1$, the Cavalieri-Lorentz weak-type theory, the Plancherel-Fourier unitary, and the product-space construction of independent random processes — six structurally distinct but Fubini-unified pillars of modern analysis.

Full proof set Master

Proposition 1 (Monotone class theorem; Halmos 1950 §6). Let $\mathcal{A}$ be an algebra of subsets of a set $\Omega$, and $\mathcal{C}$ a monotone class containing $\mathcal{A}$. Then $\mathcal{C}$ contains the $\sigma$-algebra generated by $\mathcal{A}$.

Proof. Let $\mathcal{M}(\mathcal{A})$ be the smallest monotone class containing $\mathcal{A}$. The claim is that $\mathcal{M}(\mathcal{A})$ is itself an algebra and a $\sigma$-algebra, hence equals $\sigma (\mathcal{A})$.

Define $\mathcal{F}(A)=\{B\in \mathcal{M}(\mathcal{A}):A\cup B,A\cap B,A\setminus B,B\setminus A\in \mathcal{M}(\mathcal{A})\}$ for each $A\in \mathcal{M}(\mathcal{A})$. The class $\mathcal{F}(A)$ is a monotone class (by closure of monotone-class operations under finite unions and intersections at the level of monotone limits). For $A\in \mathcal{A}$, the class $\mathcal{F}(A)$ contains $\mathcal{A}$ (since $\mathcal{A}$ is an algebra), so $\mathcal{F}(A)\supseteq \mathcal{M}(\mathcal{A})$. This says: for $A\in \mathcal{A}$ and $B\in \mathcal{M}(\mathcal{A})$, the four set-operations land in $\mathcal{M}(\mathcal{A})$. By symmetry, for $B\in \mathcal{M}(\mathcal{A})$, $\mathcal{F}(B)\supseteq \mathcal{A}$, hence $\mathcal{F}(B)=\mathcal{M}(\mathcal{A})$. So $\mathcal{M}(\mathcal{A})$ is closed under finite unions, finite intersections, and complements — an algebra.

Monotonically-closed-plus-algebraic implies $\sigma$-algebraic: a countable union ${\bigcup }_n{A}_n$ equals the monotone limit ${\bigcup }_{n\le N}{A}_n\uparrow {\bigcup }_n{A}_n$, which is in $\mathcal{M}(\mathcal{A})$ since each finite union is and monotone-class closure handles the limit. Hence $\mathcal{M}(\mathcal{A})$ is a $\sigma$-algebra containing $\mathcal{A}$, so $\mathcal{M}(\mathcal{A})\supseteq \sigma (\mathcal{A})$. Conversely $\sigma (\mathcal{A})\supseteq \mathcal{M}(\mathcal{A})$ since $\sigma$-algebras are monotone classes. Hence equality. Since the original $\mathcal{C}$ contains $\mathcal{A}$ and is monotone, $\mathcal{C}\supseteq \mathcal{M}(\mathcal{A})=\sigma (\mathcal{A})$.

$\square$

Proposition 2 (Carathéodory extension theorem; Carathéodory 1918 Vorlesungen über reelle Funktionen §5). A $\sigma$-additive function $\rho$ on an algebra $\mathcal{A}$ of subsets of $\Omega$ extends uniquely to a measure on $\sigma (\mathcal{A})$, provided $\rho$ is $\sigma$-finite on $\mathcal{A}$.

Proof sketch. Define the outer measure ${\rho }^{\ast }(E)=\inf \{{\sum }_n\rho (A_n):E\subseteq {\bigcup }_n{A}_n,A_n\in \mathcal{A}\}$. Carathéodory's outer-measure-to-measure construction (see 02.07.02) gives a $\sigma$-algebra ${\mathcal{M}}^{\ast }$ of ${\rho }^{\ast }$-measurable sets, on which ${\rho }^{\ast }$ restricts to a measure. The countable subadditivity and additivity of $\rho$ on $\mathcal{A}$ imply $\mathcal{A}\subseteq {\mathcal{M}}^{\ast }$ and ${\rho }^{\ast }{|}_{\mathcal{A}}=\rho$. Hence ${\rho }^{\ast }$ on $\sigma (\mathcal{A})\subseteq {\mathcal{M}}^{\ast }$ is the desired measure extension. Uniqueness on $\sigma (\mathcal{A})$ under $\sigma$-finiteness follows from the monotone class theorem (Proposition 1) applied to the class of sets where the candidate extensions agree.

$\square$

Proposition 3 (Product measure construction). The product measure $\mu \otimes \nu$ on $\mathcal{M}\otimes \mathcal{N}$ exists and is unique under the $\sigma$-finite hypothesis.

Proof. The measurable rectangles $\{A\times B:A\in \mathcal{M},B\in \mathcal{N}\}$ form a semiring. The set-function $\rho (A\times B):=\mu (A)\nu (B)$ extends additively to the algebra ${\mathcal{A}}_0$ of finite disjoint unions of measurable rectangles: if $E={⨆}_{i=1}^k{A}_i\times B_i$, define $\rho (E)={\sum }_i\mu (A_i)\nu (B_i)$. The well-definedness (independence of the disjoint-rectangle decomposition) and $\sigma$-additivity on ${\mathcal{A}}_0$ are proved via the section formula: for $E={⨆}_{i=1}^k{A}_i\times B_i$, ${\chi }_E(x,y)={\sum }_i{\chi }_{A_i}(x){\chi }_{B_i}(y)$, and integrating in $y$ first gives ${\int }_Y{\chi }_E(x,y)d\nu (y)={\sum }_i{\chi }_{A_i}(x)\nu (B_i)$. Integrating in $x$ gives ${\sum }_i\mu (A_i)\nu (B_i)=\rho (E)$ uniquely. Countable additivity follows by monotone convergence applied to the partial-sum sequence.

Apply Carathéodory (Proposition 2) to extend $\rho$ from ${\mathcal{A}}_0$ to a measure on $\sigma ({\mathcal{A}}_0)=\mathcal{M}\otimes \mathcal{N}$. The $\sigma$-finiteness of $\mu ,\nu$ implies $\sigma$-finiteness of $\rho$ on ${\mathcal{A}}_0$ (decompose $X=\bigcup X_k$ with $\mu (X_k)<\infty$ and $Y=\bigcup Y_{\ell }$ with $\nu (Y_{\ell })<\infty$; then $X_k\times Y_{\ell }$ has $\rho$-measure $\mu (X_k)\nu (Y_{\ell })<\infty$ and these cover $X\times Y$). Carathéodory's uniqueness gives the unique extension $\mu \otimes \nu$.

$\square$

Proposition 4 (Tonelli — full proof via standard machine). For non-negative $\mathcal{M}\otimes \mathcal{N}$-measurable $f$ and $\sigma$-finite $\mu ,\nu$, the iterated and product integrals coincide.

Proof. As detailed in Steps 1-4 of the proof of Theorem (Intermediate+ section). The four-step sequence — rectangles, monotone class theorem on indicators, linearity on simple functions, MCT on non-negative measurable functions — is the standard machine of measure theory, with the monotone class theorem (Proposition 1) doing the load-bearing extension from rectangles to general measurable sets, and MCT 02.07.04 handling the simple-to-measurable limit.

$\square$

Proposition 5 (Fubini — from Tonelli to $L^1$). For $f\in L^1(\mu \otimes \nu )$ on a $\sigma$-finite product space, the iterated and product integrals of $f$ coincide.

Proof. Decompose $f=f^{+}-f^{-}$ into non-negative real and (when complex) imaginary parts. By Tonelli applied to $|f|=f^{+}+f^{-}$ (or the four-part decomposition for complex $f$), the iterated integrals of $|f|$ all equal $\int |f|d(\mu \otimes \nu )<\infty$. Hence each of $f^{+},f^{-}$ has finite iterated integrals, and the inner integrals ${\int }_Y{f}_x^{+}d\nu ,{\int }_Y{f}_x^{-}d\nu$ are finite for $\mu$-a.e. $x$ (the set where either is infinite has $\mu$-measure zero, by the finiteness of ${\int }_X{\int }_Y{f}^{\pm }d\nu d\mu$).

On the full-measure set of good $x$'s, $f_x=f_x^{+}-f_x^{-}$ as $L^1(\nu )$-functions, and ${\int }_Y{f}_{x}d\nu ={\int }_Y{f}_x^{+}d\nu -{\int }_Y{f}_x^{-}d\nu$, a difference of two finite quantities. The function $x\mapsto {\int }_Y{f}_{x}d\nu$ is measurable (difference of two measurable functions on a full-measure set, with arbitrary measurable extension to the null set), and integrable: $${\int }_X\left|{\int }_Y{f}_{x}d\nu \right|d\mu \le {\int }_X{\int }_Y|f_x|d\nu d\mu ={\int }_{X\times Y}|f|d(\mu \otimes \nu )<\infty .$$ Linearity of the integral and the Tonelli identity for $f^{\pm }$ give ${\int }_X{\int }_Y{f}_{x}d\nu d\mu ={\int }_X(\int f_x^{+}-\int f_x^{-})d\mu =\int f^{+}d(\mu \otimes \nu )-\int f^{-}d(\mu \otimes \nu )=\int fd(\mu \otimes \nu )$. The reverse iteration is symmetric.

$\square$

Proposition 6 (Cavalieri / layer-cake from Tonelli). For non-negative measurable $f$ on $(X,\mathcal{M},\mu )$ and $p>0$: $${\int }_X{f}^{p}d\mu =p{\int }_0^{\infty }{t}^{p-1}\mu (\{f>t\})dt.$$

Proof. Write $f(x{)}^p={\int }_0^{f(x)}p{t}^{p-1}dt={\int }_0^{\infty }p{t}^{p-1}{\chi }_{\{t<f(x)\}}dt$. Tonelli on the non-negative function $(x,t)\mapsto p{t}^{p-1}{\chi }_{\{t<f(x)\}}$ on $X\times (0,\infty )$ (both spaces with their natural $\sigma$-finite measures) gives $${\int }_X{f}^{p}d\mu ={\int }_X{\int }_0^{\infty }p{t}^{p-1}{\chi }_{\{t<f(x)\}}dtd\mu (x)={\int }_0^{\infty }p{t}^{p-1}{\int }_X{\chi }_{\{f(x)>t\}}d\mu dt={\int }_0^{\infty }p{t}^{p-1}\mu (\{f>t\})dt.$$ The Cavalieri principle is a one-line Tonelli calculation.

$\square$

Proposition 7 (Convolution Banach-algebra structure on $L^1$). For $f,g\in L^1({\mathbb{R}}^n)$ with Lebesgue measure, the convolution $(f\ast g)(x)=\int f(x-y)g(y)dy$ is defined for a.e. $x$, belongs to $L^1$, and satisfies $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$.

Proof. As in Exercise 5. Tonelli on $|f(x-y)||g(y)|$ plus translation-invariance of Lebesgue measure gives the integral on ${\mathbb{R}}^n\times {\mathbb{R}}^n$ equal to $\Vert f{\Vert }_1\Vert g{\Vert }_1$. Hence the inner $y$-integral is finite for a.e. $x$, defining $(f\ast g)(x)$ as an absolutely convergent integral on a full-measure set. The $L^1$-norm bound follows from $|(f\ast g)(x)|\le \int |f(x-y)||g(y)|dy$ and integration in $x$.

$\square$

Proposition 8 (Sierpiński's non-$\sigma$-finite counterexample). On $X=Y=[0,1]$ with $\mu$ Lebesgue and $\nu$ counting measure, the two iterated integrals of ${\chi }_{\Delta }$ (diagonal indicator) differ.

Proof. Each $\nu$-section $({\chi }_{\Delta }{)}_x={\chi }_{\{x\}}$ has ${\int }_Y{\chi }_{\{x\}}d\nu =\nu (\{x\})=1$ (counting measure of a singleton). The outer Lebesgue integral ${\int }_{X}1dm=1$. Each $\mu$-section $({\chi }_{\Delta }{)}^y={\chi }_{\{y\}}$ has ${\int }_X{\chi }_{\{y\}}dm=m(\{y\})=0$ (Lebesgue measure of a singleton). The outer counting integral ${\int }_{Y}0d\nu =0$. The iterated integrals are $1$ and $0$ — unequal, and Fubini-Tonelli fails because $\nu$ is not $\sigma$-finite on $[0,1]$ (any decomposition of the uncountable set $[0,1]$ into measurable pieces of finite counting measure would require uncountably many pieces).

$\square$

Proposition 9 (Completion-compatible Fubini-Tonelli; Folland 2e Theorem 2.39). With $\overline{\mu \otimes \nu }$ the completion of the product, the Fubini-Tonelli identity holds for $\overline{\mu \otimes \nu }$-measurable functions, with section-measurability holding $\mu$-a.e. (rather than for all $x$).

Proof sketch. For $E\in \overline{\mathcal{M}\otimes \mathcal{N}}$, decompose $E=E^{\prime }\cup N$ with $E^{\prime }\in \mathcal{M}\otimes \mathcal{N}$ and $N\subseteq M$ for some $M\in \mathcal{M}\otimes \mathcal{N}$ with $(\mu \otimes \nu )(M)=0$. By Tonelli for $M$, ${\int }_X\nu (M_x)d\mu =0$, so $\nu (M_x)=0$ for $\mu$-a.e. $x$. On this full-measure set, $E_x=E_x^{\prime }\cup N_x$ with $\nu (N_x)\le \nu (M_x)=0$, so $E_x$ is $\overline{\nu }$-measurable (containing the $\mathcal{N}$-measurable $E_x^{\prime }$ up to a $\nu$-null set). The iterated-integral identity for ${\chi }_E$ reduces to the identity for ${\chi }_{E^{\prime }}$ via the a.e.-zero null-set correction, and the standard machine carries this through to general $\overline{\mu \otimes \nu }$-measurable functions.

$\square$

Proposition 10 (Independence and product measures in probability). Let $X_1,\dots ,X_n$ be random variables on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with values in measurable spaces $(E_i,{\mathcal{E}}_i)$. The variables are independent if and only if the joint distribution $\mathbb{P}\circ (X_1,\dots ,X_n{)}^{-1}$ on ${\prod }_i{E}_i$ equals the product measure ${⨂}_i(\mathbb{P}\circ X_i^{-1})$ of the marginal distributions.

Proof. (Forward.) Independence means $\mathbb{P}(X_1\in A_1,\dots ,X_n\in A_n)={\prod }_i\mathbb{P}(X_i\in A_i)$ for all $A_i\in {\mathcal{E}}_i$. Translating: the joint distribution on the rectangle $A_1\times \cdots \times A_n$ equals the product of marginal-distribution masses, which is exactly the rectangle-formula defining the product measure. By Theorem 2 (uniqueness on rectangles forces equality on the $\sigma$-algebra), the joint distribution equals the product measure.

(Reverse.) If the joint distribution is the product measure, the rectangle formula holds, which is the independence condition.

Fubini-Tonelli applied to the product measure gives the expectation-factorisation identity for independent integrable $f_i(X_i)$: $\mathbb{E}[{\prod }_i{f}_i(X_i)]={\prod }_i\mathbb{E}[f_i(X_i)]$, the foundational identity behind the laws of large numbers and the central limit theorem.

$\square$

Connections Master

• Lebesgue integral and monotone convergence 02.07.04. The direct prerequisite for Tonelli: MCT applied to the simple-function approximations $s_n\uparrow f$ in Step 4 of the Tonelli proof gives the iterated-integral identity for general non-negative measurable $f$ from its truth on simple functions. Without MCT the standard machine breaks at the simple-to-general step.

• Fatou's lemma and dominated convergence 02.07.05. The complementary tool. DCT is used in two structural places: (i) the monotone-decreasing sub-step of the monotone-class argument for Tonelli (Sub-step 2c), requiring $\sigma$-finiteness for the dominator; and (ii) Fubini's reduction from non-negative to $L^1$, where the iterated integral of the negative part $f^{-}$ is controlled by the DCT-style identity $\int f=\int f^{+}-\int f^{-}$ valid only under absolute integrability.

• $L^p$ spaces and Riesz-Fischer completeness 02.07.06. The just-shipped peer. Fubini-Tonelli is the foundational fact behind the convolution Banach-algebra structure on $L^1({\mathbb{R}}^n)$ (Theorem 8 / Proposition 7): the $L^1\times L^1\to L^1$ convolution boundedness is a one-line Tonelli computation. The same Tonelli interchange underlies the boundedness of all the Young-inequality convolution maps $L^p\times L^q\to L^r$ with $1/p+1/q=1+1/r$, by Riesz-Thorin interpolation between Fubini-Tonelli endpoints.

• Lebesgue outer measure and Carathéodory construction 02.07.02. The construction of the product measure $\mu \otimes \nu$ on $\mathcal{M}\otimes \mathcal{N}$ relies on Carathéodory's extension theorem (Proposition 2) applied to the additive set-function $\mu \otimes \nu$ on the algebra of finite disjoint unions of measurable rectangles. Without Carathéodory's outer-measure machinery the product measure has no $\sigma$-additive existence statement.

• Fourier transform [forward: 02.10.04 Fourier-Plancherel]. The Fubini-Tonelli identity is invoked in essentially every Fourier-analytic computation: the Plancherel identity, the Fourier-inversion formula, the convolution-product identity $\widehat{f\ast g}=\hat{f}\cdot \hat{g}$, and the Hausdorff-Young inequality all rely on Fubini interchanges. The forward connection to Fourier analysis is the load-bearing application of Fubini-Tonelli in harmonic analysis.

• Convolution algebras [forward: convolution Banach algebra of $L^1(G)$ on locally compact groups]. Proposition 7 generalises: for a locally compact group $G$ with Haar measure $m$, $L^1(G,m)$ is a Banach algebra under convolution, and the Fubini-Tonelli identity is the foundational fact identifying convolution as a bounded $L^1\times L^1\to L^1$ operation. This is the starting point of abstract harmonic analysis (Hewitt-Ross 1963 Abstract Harmonic Analysis).

• Sobolev embeddings [forward: 02.13.x Sobolev spaces]. Fubini-Tonelli is the foundational fact behind Gagliardo-Nirenberg-Sobolev inequalities, the iterated-slicing inequality $\Vert f{\Vert }_{n/(n-1)}\le C{\prod }_{i=1}^n\Vert {\partial }_{i}f{\Vert }_1^{1/n}$ for compactly supported $f$ on ${\mathbb{R}}^n$. The slicing argument is a Tonelli-style interchange in each coordinate, with the product structure of ${\mathbb{R}}^n$ entering through the product-measure-and-Lebesgue identification.

• Probability product spaces [lateral: probability theory]. Proposition 10 identifies independent random variables with joint distributions that are product measures, and the Fubini-Tonelli expectation-factorisation identity $\mathbb{E}[fg]=\mathbb{E}[f]\mathbb{E}[g]$ for independent integrable $f,g$ is the foundational identity of probability theory. The infinite-product Kolmogorov extension theorem extends the finite-product construction to infinite product spaces, the foundation for stochastic processes and Wiener measure.

Historical & philosophical context Master

Cavalieri's 1635 Geometria indivisibilibus continuorum nova quadam ratione promota ^{[Cavalieri 1635]} was the seventeenth-century forerunner of the iterated-integral interchange: the area under a curve is the sum of the lengths of vertical slices, and Cavalieri's principle that two solids with equal cross-section areas at every height have equal volumes is the geometric germ of Tonelli's theorem. Cavalieri's contemporary Torricelli (1647) extended the principle to derive volumes of paraboloid and conoid solids, and Pascal (1654) used it to compute areas under cycloid curves. The principle was made rigorous only after the Lebesgue integral was developed in the early twentieth century.

Cauchy's 1814 Mémoire sur les intégrales définies introduced the first general statement of the iterated-integral interchange for continuous functions on rectangles, using uniform convergence of the inner integral as a function of the outer parameter. Riemann's 1854 Habilitationsschrift refined the statement to Riemann-integrable functions, with the famous condition that the function be Riemann-integrable on the rectangle (a more restrictive condition than Lebesgue-integrability, but already covering the practical applications of nineteenth-century analysis).

Henri Lebesgue's 1902 Intégrale, longueur, aire (his thesis) introduced the Lebesgue integral and stated an iterated-integral interchange for bounded measurable functions on bounded rectangles. The general statement for unbounded functions and infinite domains was incomplete in Lebesgue's 1902 work; the missing piece was the careful treatment of $\sigma$-finiteness and the absolute-integrability condition.

Guido Fubini's 1907 Rendiconti dell'Accademia dei Lincei paper ^{[Fubini 1907]}, titled Sugli integrali multipli (On multiple integrals), gave the first general statement of the iterated-integral interchange for absolutely integrable Lebesgue-measurable functions on $\sigma$-finite product spaces, with the proof via the positive/negative decomposition $f=f^{+}-f^{-}$ and reduction to the non-negative case. Fubini's statement was restricted to bounded rectangles in ${\mathbb{R}}^m\times {\mathbb{R}}^n$, with general $\sigma$-finite spaces appearing only after Carathéodory's 1918 Vorlesungen über reelle Funktionen axiomatised the measure-theoretic framework.

Leonida Tonelli's 1909 Rendiconti dell'Accademia dei Lincei paper ^{[Tonelli 1909]}, titled Sull'integrazione per parti (On integration by parts), gave the non-negative version of the iterated-integral identity. Tonelli's recognition that absolute integrability was unnecessary for non-negative measurable functions — both iterated integrals always coincide with the double integral as elements of $[0,\infty ]$, with all three possibly being $+\infty$ simultaneously — completed the modern Fubini-Tonelli framework. Tonelli's paper appeared two years after Fubini's, and the combined Fubini-Tonelli identity (apply Tonelli to $|f|$ for the integrability test, then Fubini to the signed $f$) became standard practice in measure theory by the 1920s.

Wacław Sierpiński's 1920 Fundamenta Mathematicae paper ^{[Sierpiński 1920]}, titled Sur un problème concernant les ensembles mesurables superficiellement (On a problem concerning superficially measurable sets), exhibited the first explicit counterexample to the iterated-integral identity in the absence of $\sigma$-finiteness: a Lebesgue-measurable function on $[0,1{]}^2$ whose two iterated integrals exist as Lebesgue integrals but disagree. Sierpiński's counterexample required the continuum hypothesis; later cleaner counterexamples (using non-$\sigma$-finite counting measure, as in Theorem 7) work in ZFC without set-theoretic auxiliary axioms.

Paul Halmos's 1950 Measure Theory monograph ^{[Halmos 1950]} codified the modern textbook formulation of Fubini-Tonelli: the monotone-class-theorem-based proof, the $\sigma$-finiteness hypothesis, the rectangle formula and Carathéodory product-measure construction, and the standard machine of indicator-to-simple-to-non-negative-to-signed extension. Halmos's textbook treatment was followed by Folland 1984 Real Analysis §2.5, Royden 1968 Real Analysis §11, Rudin 1966 Real and Complex Analysis §8, and Bogachev 2007 Measure Theory Vol. 1 §3.3-3.4 ^{[Bogachev Volume 1; Folland 2e; Rudin Real and Complex Analysis]}.

The philosophical thread: Fubini-Tonelli is the rigorous version of the seventeenth-century Cavalieri principle, with three centuries of mathematical refinement adding (a) the Lebesgue-integral machinery permitting unbounded and singular integrands, (b) the $\sigma$-finiteness condition delineating where the interchange is permitted, (c) the absolute-integrability versus non-negativity distinction (Fubini vs. Tonelli), and (d) the explicit Sierpiński counterexamples showing the necessity of the hypotheses. The arc from Cavalieri 1635 to Halmos 1950 spans 315 years and tracks the maturation of integration theory from geometric intuition to a fully axiomatic theorem.

Bibliography Master

@book{Cavalieri1635,
  author    = {Cavalieri, Bonaventura},
  title     = {Geometria indivisibilibus continuorum nova quadam ratione promota},
  publisher = {Typis Clementis Ferronij},
  address   = {Bologna},
  year      = {1635}
}

@article{Fubini1907,
  author  = {Fubini, Guido},
  title   = {Sugli integrali multipli},
  journal = {Rendiconti della Reale Accademia dei Lincei},
  volume  = {16},
  year    = {1907},
  pages   = {608--614}
}

@article{Tonelli1909,
  author  = {Tonelli, Leonida},
  title   = {Sull'integrazione per parti},
  journal = {Rendiconti della Reale Accademia dei Lincei},
  volume  = {18},
  year    = {1909},
  pages   = {246--253}
}

@article{Fubini1915,
  author  = {Fubini, Guido},
  title   = {Sulla derivazione per serie},
  journal = {Atti della Reale Accademia dei Lincei. Classe di Scienze Fisiche},
  series  = {5},
  volume  = {24},
  year    = {1915},
  pages   = {204--206}
}

@book{Caratheodory1918,
  author    = {Carath\'eodory, Constantin},
  title     = {Vorlesungen \"uber reelle {F}unktionen},
  publisher = {Teubner},
  address   = {Leipzig and Berlin},
  year      = {1918}
}

@article{Sierpinski1920,
  author  = {Sierpi\'nski, Wac\l aw},
  title   = {Sur un probl\`eme concernant les ensembles mesurables superficiellement},
  journal = {Fundamenta Mathematicae},
  volume  = {1},
  year    = {1920},
  pages   = {112--115}
}

@article{Young1912,
  author  = {Young, William Henry},
  title   = {On the multiplication of successions of {F}ourier constants},
  journal = {Proceedings of the London Mathematical Society},
  series  = {2},
  volume  = {11},
  year    = {1912},
  pages   = {357--366}
}

@book{Halmos1950,
  author    = {Halmos, Paul R.},
  title     = {Measure Theory},
  publisher = {Van Nostrand},
  address   = {New York},
  year      = {1950}
}

@book{Folland1999,
  author    = {Folland, Gerald B.},
  title     = {Real Analysis: Modern Techniques and Their Applications},
  edition   = {2},
  publisher = {Wiley},
  year      = {1999}
}

@book{Rudin1987,
  author    = {Rudin, Walter},
  title     = {Real and Complex Analysis},
  edition   = {3},
  publisher = {McGraw-Hill},
  year      = {1987}
}

@book{Bogachev2007,
  author    = {Bogachev, Vladimir I.},
  title     = {Measure Theory, Volume 1},
  publisher = {Springer},
  year      = {2007}
}

@book{RoydenFitzpatrick2010,
  author    = {Royden, H. L. and Fitzpatrick, P. M.},
  title     = {Real Analysis},
  edition   = {4},
  publisher = {Pearson},
  year      = {2010}
}

@book{Saks1937,
  author    = {Saks, Stanis\l aw},
  title     = {Theory of the Integral},
  edition   = {2},
  publisher = {Hafner},
  address   = {New York},
  year      = {1937}
}

@book{Tao2016,
  author    = {Tao, Terence},
  title     = {Analysis II},
  edition   = {3},
  publisher = {Hindustan Book Agency},
  year      = {2016}
}

@article{BrascampLieb1976,
  author  = {Brascamp, Herm Jan and Lieb, Elliott H.},
  title   = {Best constants in {Y}oung's inequality, its converse, and its generalization to more than three functions},
  journal = {Advances in Mathematics},
  volume  = {20},
  year    = {1976},
  pages   = {151--173}
}

@book{HewittRoss1963,
  author    = {Hewitt, Edwin and Ross, Kenneth A.},
  title     = {Abstract Harmonic Analysis, Volume I},
  publisher = {Springer},
  year      = {1963}
}