Fubini’S Theorem

1. Core Concept Overview & Intuition

Fubini’s Theorem is the central result that permits the interchange of the order of integration in a multiple integral. In the setting of product measure spaces, it asserts that integrating a measurable function over the product space $U_1 \times U_2$ can be accomplished by iterated integration: first with respect to one coordinate, then with respect to the other, and that the two possible orders yield the same value.

The theorem rests on three logically distinct layers. The first is a measure-zero criterion: a subset of the product space has measure zero precisely when almost every section has measure zero. The second is a non-negativity layer: for non-negative measurable functions, the iterated integrals always exist (possibly infinite) and coincide with the product integral. The third is an integrability layer: when the function is integrable, almost every section is integrable, the section integrals define integrable functions on each factor, and the iterated integrals reproduce the product integral.

The intuition is that a double sum over a grid can be computed row-by-row or column-by-column, and the total is the same. Fubini’s Theorem is the continuous, measure-theoretic analogue of this finite double-sum identity. The measure-zero criterion is the technical device that makes the passage from sections to the whole product rigorous: it guarantees that exceptional sections, where behaviour might fail, can be discarded without affecting the integral.

The theorem is intimately connected with the convergence tools developed earlier in the course. The dominated convergence theorem is used to justify limiting arguments on sections, and the closure properties of convergence in probability (scalar multiples, sums, products, quotients) supply the algebraic stability that underlies the manipulation of measurable functions on product spaces.

2. Mathematical Formalization & Core Definitions

2.1 Product space and sections

Let $(U_1, \mathcal{A}_1, \mu_1)$ and $(U_2, \mathcal{A}_2, \mu_2)$ be measure spaces, and let $\lambda$ denote the product measure on the product $\sigma$-field of $U_1 \times U_2$. For a measurable set $A \subseteq U_1 \times U_2$ and a point $w_1 \in U_1$, the $w_1$-section of $A$ is the set of second coordinates $w_2$ such that $(w_1, w_2) \in A$. The $w_2$-section is defined analogously by fixing the second coordinate. Sections of a measurable function $h$ on $U_1 \times U_2$ are obtained by fixing one coordinate and viewing the resulting function of the remaining coordinate.

2.2 Almost-everywhere statements

A property is said to hold $\mu_1$-almost everywhere if the set of points where it fails has $\mu_1$-measure zero. The phrase “$\mu_1$-almost every $w_1$-section” means that the property holds for all $w_1$ outside a $\mu_1$-null set. This terminology is essential for stating the measure-zero criterion precisely.

2.3 Measurable functions on the product space

A function $h$ on $U_1 \times U_2$ is measurable if it is measurable with respect to the product $\sigma$-field. Non-negative measurable functions are permitted to take the value $+\infty$, and their integrals are defined in the extended sense. Integrable functions are those measurable functions whose integral of the absolute value is finite.

2.4 Iterated integrals

For a non-negative measurable function $h$ on $U_1 \times U_2$, the iterated integrals are written as

$$
\iint h \, d\mu_1 \, d\mu_2
$$

and

$$
\iint h \, d\mu_2 \, d\mu_1.
$$

The first symbol denotes integration with respect to $\mu_2$ first (for each fixed $w_1$), followed by integration with respect to $\mu_1$. The second denotes the reverse order. The product integral is written $\int h \, d\lambda$.

2.5 Section integrals

When $h$ is integrable on $U_1 \times U_2$, the section integrals are defined by

$$
f(w_1) = \int h \, d\mu_2 (w_2)
$$

and

$$
g(w_2) = \int h \, d\mu_1 (w_1).
$$

Here $f$ is a function on $U_1$ obtained by integrating out the second variable, and $g$ is a function on $U_2$ obtained by integrating out the first variable.

2.6 Closure under arithmetic operations for convergence in probability

The algebraic stability of convergence in probability is a prerequisite for manipulating measurable functions in limiting arguments. The following result is stated for sequences of random variables $X_n$ and $Y_n$ with $X_n \to X$ and $Y_n \to Y$ in probability, and a real constant $a$.

> Theorem (Closure under arithmetic operations). Let $X_n \to X$ and $Y_n \to Y$ in probability and let $a \in \mathbb{R}$. Then:
> 1. $aX_n \to aX$ in probability (scalar multiple);
> 2. $X_n + Y_n \to X + Y$ in probability (additive);
> 3. $X_n Y_n \to XY$ in probability (multiplicative);
> 4. $X_n / Y_n \to X / Y$ in probability (quotient), where $P(Y_n = 0) = 0$ and $P(Y = 0) = 0$.

The proof of the scalar-multiple case reduces to the observation that for $\epsilon > 0$,

$$
P\left[\lvert aX_n \rvert > \epsilon\right] = P\left[\lvert a \rvert \lvert X_n \rvert > \epsilon\right] = P\left[\lvert X_n \rvert > \frac{\epsilon}{\lvert a \rvert}\right] \to 0
$$

as $n \to \infty$, with the case $a = 0$ being trivial. The additive case uses the triangle inequality and a splitting of the threshold:

$$
P\left[\lvert X_n + Y_n - X - Y \rvert > \epsilon\right] \le P\left[\lvert X_n - X \rvert + \lvert Y_n - Y \rvert > \epsilon\right] \le P\left[\lvert X_n - X \rvert > \frac{\epsilon}{2}\right] + P\left[\lvert Y_n - Y \rvert > \frac{\epsilon}{2}\right].
$$

Both terms on the right tend to zero. The multiplicative case inserts and subtracts the cross term $X_n Y$:

$$
P\left[\lvert X_n Y_n - XY \rvert > \epsilon\right] = P\left[\lvert X_n(Y_n - Y) + Y(X_n - X) \rvert > \epsilon\right] \le P\left[\lvert X_n(Y_n - Y) \rvert > \frac{\epsilon}{2}\right] + P\left[\lvert Y(X_n - X) \rvert > \frac{\epsilon}{2}\right].
$$

The first term is controlled by a truncation argument: for any $\sigma > 0$ and $\eta = \epsilon / (2\sigma) > 0$,

$$
P\left[\lvert X_n(Y_n - Y) \rvert > \frac{\epsilon}{2}\right] \le P\left[\lvert X_n - X \rvert > \eta\right] + P\left[\lvert Y_n - Y \rvert > \sigma\right] + P\left[\lvert X \rvert > \eta\right],
$$

where the first two terms tend to zero and the last can be made arbitrarily small by taking $\eta$ large, since $X$ is finite almost surely. The second term vanishes because $\lvert Y \rvert < \infty$ almost surely and $X_n \to X$ in probability. The quotient case follows the same pattern after writing

$$
\frac{X_n}{Y_n} - \frac{X}{Y} = \frac{(X_n - X)Y + X(Y - Y_n)}{Y_n Y},
$$

using $P(Y_n = 0) = P(Y = 0) = 0$.