Fubini-Tonelli theorem and the $\sigma$-finiteness hypothesis

Let $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ be $\sigma$-finite measure spaces and let $f$ be measurable for the product $\sigma$-algebra $\mathcal{A} \otimes \mathcal{B}$.

Tonelli. If $f \ge 0$ then $x \mapsto \int_Y f(x,y)\, d\nu(y)$ and $y \mapsto \int_X f(x,y) \, d\mu(x)$ are measurable and

$\int_X \! \Big( \int_Y f \, d\nu \Big) d\mu \;=\; \int_Y \! \Big( \int_X f \, d\mu \Big) d\nu \;=\; \int_{X \times Y} f \, d(\mu \otimes \nu),$

all three possibly $+\infty$.

Fubini. If $f \in L^{1}(\mu \otimes \nu)$ then $f(x, \cdot) \in L^1(\nu)$ for $\mu$-almost every $x$, $f(\cdot, y) \in L^1(\mu)$ for $\nu$-almost every $y$, the two almost everywhere defined iterated integrals are integrable, and the same chain of equalities holds.

In practice the two are used together: Tonelli applied to $|f|$ establishes the integrability hypothesis of Fubini, which is then applied to $f$. Neither theorem asserts that equality of the two iterated integrals implies integrability, and neither dispenses with $\sigma$-finiteness.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. Construction of the product measure by Caratheodory extension from measurable rectangles, uniqueness of that extension on $\sigma$-finite spaces by a Dynkin system argument, the fact that sections of a product-measurable set are measurable, and then the monotone convergence theorem (Monotone convergence theorem (Beppo Levi) ‡) to pass from indicators to simple functions to nonnegative measurable functions. $\sigma$-finiteness is used twice: once to make the product measure unique, once to make the section function measurable.

Which page it serves. The Fubini and change of variables page of the multivariable track. That page proves the Fubini theorem for the Riemann integral on a box, where the hypothesis is continuity or Riemann integrability of $f$ together with existence of the inner integrals, and where the counterexamples are about existence rather than about measurability. The Lebesgue version is what makes the theorem usable for the functions that actually arise, and its two failure modes are recorded here as Failure of Tonelli without $\sigma$-finiteness: the diagonal under Lebesgue times counting measure ‡ and Sierpinski's example under the continuum hypothesis ‡.

Two hypotheses that are often dropped and should not be. The first is $\sigma$-finiteness, whose failure is the diagonal example. The second is product measurability of $f$: equality of iterated integrals for a function that is not measurable on the product is not asserted by anything above, and Sierpinski's example shows the iterated integrals can then exist and differ.