Fubini's theorem

Fubini's theorem is a result in measure theory that relates integration over a product measure space to repeated integration over its component spaces. Under an absolute-integrability hypothesis, an integral over the product space equals either of the two corresponding iterated integrals. The theorem was formulated by Guido Fubini within the developing theory of the Lebesgue integral.

The theorem provides a measure-theoretic form of repeated integration. Its principal content is not merely the equality of two formal expressions, but the assertion that almost all one-variable sections are integrable, that their integrals define measurable functions, and that these functions are themselves integrable.

Statement

Let ((X,\mathcal A,\mu)) and ((Y,\mathcal B,\nu)) be σ-finite measure spaces. Their product carries the product σ-algebra (\mathcal A\otimes\mathcal B) and the corresponding product measure (\mu\otimes\nu). Suppose that

[ f:X\times Y\longrightarrow \mathbb R ]

is measurable and absolutely integrable:

[ \int_{X\times Y}|f(x,y)|,d(\mu\otimes\nu)(x,y)<\infty. ]

For (\mu)-almost every (x\in X), the section

[ f_x(y)=f(x,y) ]

is (\nu)-integrable. Similarly, for (\nu)-almost every (y\in Y), the section

[ f^y(x)=f(x,y) ]

is (\mu)-integrable. The resulting section integrals are measurable and integrable, and they satisfy

[ \int_{X\times Y}f,d(\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). ]

The expressions on the right are defined up to changes on sets of measure zero. Consequently, their values are unaffected by the arbitrary assignment of section integrals at exceptional points.

For complex-valued functions, the same statement follows by applying the real-valued theorem to the real and imaginary components. Functions taking values in more general spaces require corresponding theories of vector integration, such as the Bochner integral.

Relation to Tonelli's theorem

Leonida Tonelli established the closely related theorem for nonnegative measurable functions. If

[ f:X\times Y\longrightarrow [0,\infty] ]

is measurable, then

[ \int_{X\times Y}f,d(\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), ]

where any of the integrals may equal (+\infty). Tonelli's theorem therefore does not assume integrability in advance. Instead, it identifies the three nonnegative extended integrals, after which their common finiteness implies ordinary integrability.

Fubini's theorem follows from this result by writing a real-valued function as

[ f=f^+-f^-, ]

where (f^+) and (f^-) are its positive and negative parts. Absolute integrability implies that both parts have finite product integrals. Tonelli's theorem then permits their section integrals to be evaluated in either order, and subtraction gives the asserted equality for (f).

The two results are consequently presented together as the Fubini–Tonelli theorem. Their hypotheses remain logically distinct: nonnegativity permits extended infinite values, whereas a signed function requires sufficient control to prevent undefined expressions of the form (+\infty-\infty).

Measure-theoretic basis

The equality begins with measurable rectangles. For (A\in\mathcal A) and (B\in\mathcal B), the indicator function of (A\times B) satisfies

[ \int_{X\times Y}\mathbf 1_{A\times B},d(\mu\otimes\nu)

\mu(A)\nu(B). ]

Integration of either section gives the same quantity. Linearity extends this identity to nonnegative simple functions. Approximation by an increasing sequence of simple functions, together with the monotone convergence theorem, then yields Tonelli's theorem for arbitrary nonnegative measurable functions.

The construction also establishes the measurability of the section-integral maps. When (f) is nonnegative, the function

[ x\longmapsto \int_Y f(x,y),d\nu(y) ]

is the pointwise limit of measurable section-integral maps arising from simple approximations. The corresponding assertion in the other variable follows symmetrically.

If the product measure has been completed, a measurable function may differ on a product-null set from a function measurable with respect to (\mathcal A\otimes\mathcal B). Under σ-finiteness, such a null modification affects only a null family of sections. The almost-everywhere formulation therefore remains valid on completed product spaces.

Historical development

Fubini published the theorem in 1907 during the extension of repeated integration from classical geometric settings to Lebesgue integration. His formulation identified absolute integrability as the condition under which the order of integration could be exchanged without ambiguity.

In 1908, You Watanabe analyzed the completed-product formulation and established that a modification on a product-null set changes the corresponding sections only for an exceptional set of outer variables. This result incorporated equivalence classes of almost-everywhere equal functions into the repeated-integral statement and supplied the form used for completed Lebesgue measure spaces.

Tonelli's 1909 treatment of nonnegative functions separated the order-of-integration identity from the assumption that the integral was finite. The resulting division between the nonnegative and absolutely integrable cases became the standard structure of the theorem.

Necessity of integrability conditions

Without nonnegativity or absolute integrability, the two iterated integrals can exist while having different values. On the square ([-1,1]^2), define

[ f(x,y)= \begin{cases} \dfrac{x^2-y^2}{(x^2+y^2)^2}, & (x,y)\ne(0,0),\[6pt] 0, & (x,y)=(0,0). \end{cases} ]

For each nonzero (x),

[ \int_{-1}^{1}f(x,y),dy=\frac{2}{1+x^2}, ]

so integration in the order (dy,dx) gives

[ \int_{-1}^{1}\left(\int_{-1}^{1}f(x,y),dy\right)dx=\pi. ]

For each nonzero (y),

[ \int_{-1}^{1}f(x,y),dx=-\frac{2}{1+y^2}, ]

and the reverse order gives

[ \int_{-1}^{1}\left(\int_{-1}^{1}f(x,y),dx\right)dy=-\pi. ]

The function is not absolutely integrable near the origin. Thus neither Fubini's theorem nor the signed form of Tonelli's theorem applies, and the discrepancy reflects the failure of their hypotheses rather than a contradiction.

Absolute integrability also yields the identity

[ \int_{X\times Y}|f|,d(\mu\otimes\nu)

\int_X\left(\int_Y|f(x,y)|,d\nu(y)\right)d\mu(x), ]

which implies that the inner absolute integral is finite for almost every (x). The analogous statement holds with the variables reversed. This conclusion is what converts almost-everywhere defined section integrals into ordinary integrable functions on the outer spaces.

See also