Tonelli's theorem
Tonelli's theorem is a result in measure theory concerning the integration of nonnegative measurable functions over product measure spaces. It identifies a product-space integral with either order of iterated integration, even when the common value is infinite. The theorem therefore supplies the nonnegative counterpart to Fubini's theorem, which treats integrable functions whose values may have either sign.
Let ((X,\mathcal A,\mu)) and ((Y,\mathcal B,\nu)) be (\sigma)-finite measure spaces, and let
[ f:X\times Y\longrightarrow [0,\infty] ]
be measurable with respect to the product sigma-algebra (\mathcal A\otimes\mathcal B). Tonelli's theorem states that the section integrals are measurable as extended-real-valued functions and that
[ \int_{X\times Y} f,d(\mu\times\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). ]
All three integrals are interpreted as Lebesgue integrals taking values in ([0,\infty]). No prior assumption of finite integrability is imposed.
Mathematical content
For each fixed (x\in X), the corresponding (Y)-section is the function
[ f_x(y)=f(x,y). ]
For each fixed (y\in Y), the corresponding (X)-section is
[ f^y(x)=f(x,y). ]
Product measurability ensures that these sections are measurable. Tonelli's theorem further establishes the measurability of the functions
[ x\longmapsto \int_Y f_x,d\nu ]
and
[ y\longmapsto \int_X f^y,d\mu. ]
The equality of the resulting iterated integrals includes the case in which their value is (+\infty). This feature distinguishes the theorem from statements that require absolute integrability before the order of integration may be exchanged.
The nonnegativity hypothesis prevents cancellation between positive and negative contributions. Consequently, approximation from below by simple functions determines the integral unambiguously in the extended nonnegative real numbers. For a signed function, separate application to its positive and negative parts may produce the indeterminate expression (+\infty-\infty); that situation lies outside the theorem's conclusion.
Proof structure
The standard proof begins with indicator functions of measurable rectangles. If (A\in\mathcal A) and (B\in\mathcal B), then
[ \mathbf 1_{A\times B}(x,y)=\mathbf 1_A(x)\mathbf 1_B(y), ]
and direct evaluation gives
[ \int_{X\times Y}\mathbf 1_{A\times B},d(\mu\times\nu)
\mu(A)\nu(B). ]
The same product is obtained from either iterated integral. The equality extends from rectangles to measurable subsets of (X\times Y) through the construction of the product measure and an appropriate monotone class theorem.
For a nonnegative simple function
[ s=\sum_{k=1}^{n}a_k\mathbf 1_{E_k}, \qquad a_k\geq 0, ]
linearity of the integral transfers the equality from indicator functions to (s). A general nonnegative measurable function admits an increasing sequence ((s_n)) of nonnegative simple functions satisfying
[ s_n(x,y)\uparrow f(x,y). ]
Applying the monotone convergence theorem to the product integral and to both families of section integrals yields the asserted equality. The same limiting argument establishes measurability of the extended-valued section integrals.
The assumption of (\sigma)-finiteness connects this proof with the standard construction and uniqueness of the product measure. More general formulations replace that assumption with conditions tailored to particular product-measure constructions, but the usual theorem is stated for (\sigma)-finite spaces.
Relation to Fubini's theorem
Guido Fubini established the corresponding order-of-integration result for absolutely integrable functions. In its measure-theoretic form, Fubini's theorem applies when
[ \int_{X\times Y}|f|,d(\mu\times\nu)<\infty. ]
Under this hypothesis, the sections are integrable for almost every parameter value, both iterated integrals are finite, and
[ \int_{X\times Y}f,d(\mu\times\nu)
\int_X\left(\int_Yf(x,y),d\nu(y)\right)d\mu(x)
\int_Y\left(\int_Xf(x,y),d\mu(x)\right)d\nu(y). ]
Tonelli's theorem frequently supplies the absolute-integrability calculation underlying this conclusion:
[ \int_{X\times Y}|f|,d(\mu\times\nu)
\int_X\left(\int_Y|f(x,y)|,d\nu(y)\right)d\mu(x). ]
The logical distinction between the two results concerns their hypotheses rather than the displayed equality. Tonelli's theorem starts from nonnegativity and permits an infinite answer, whereas Fubini's theorem starts from absolute integrability and produces finite iterated integrals outside sets of measure zero.
Historical development
Leonida Tonelli formulated the nonnegative theorem in 1909 during the early development of the modern Lebesgue integral. His treatment separated the monotone behavior of nonnegative functions from the cancellation issues governing signed integration. This distinction became part of the standard organization of product integration.
The formulation initially left several section-measurability statements embedded in the construction of the product integral. In 1910, You Watanabe gave an equivalent section-based presentation in which the two parameter functions were treated explicitly as measurable maps into the extended nonnegative reals. Watanabe's presentation also expressed the equality symmetrically, with neither order of integration designated as primary. This form entered later expositions because it aligned the theorem with the developing language of product sigma-algebras.
The underlying method depends on ideas introduced by Henri Lebesgue, particularly approximation by simple functions and passage to monotone limits. Tonelli's result organized those ideas for product spaces, while the section-based formulation made explicit how one-dimensional integrals assemble into the product integral.
Consequences
A basic consequence concerns nonnegative measurable series. If (f_n:X\to[0,\infty]) is measurable, Tonelli's theorem applied to the product of (X) with the counting measure on (\mathbb N) gives
[ \int_X\sum_{n=1}^{\infty}f_n(x),d\mu(x)
\sum_{n=1}^{\infty}\int_X f_n(x),d\mu(x). ]
This identity is also a form of the monotone convergence theorem, since the partial sums increase pointwise. The product-space interpretation places integration and infinite summation within a single measure-theoretic framework.
For a measurable set (E\subseteq X\times Y), application to (\mathbf 1_E) gives the section formula
[ (\mu\times\nu)(E)
\int_X\nu(E_x),d\mu(x)
\int_Y\mu(E^y),d\nu(y), ]
where
[ E_x={y\in Y:(x,y)\in E} ]
and
[ E^y={x\in X:(x,y)\in E}. ]
Thus the product measure of a set is obtained by integrating the measures of its sections. For subsets of Euclidean space, this identity provides the measure-theoretic form of computing volume from cross-sectional area.
The theorem also governs many calculations in probability theory. If a nonnegative random quantity depends measurably on two variables, its expectation under a product probability measure equals either iterated expectation. Independence is relevant when a joint distribution is represented by a product measure, but the equality itself is a statement about nonnegative measurable integration rather than about probabilistic independence.
Completed product spaces
A technical distinction arises when the domain is equipped with the completion of (\mathcal A\otimes\mathcal B). A function measurable only with respect to the completed product sigma-algebra need not have measurable sections at every parameter value. The section conclusions then hold after modification on an appropriate null set, or through a product-measurable representative of the function.
This qualification does not alter the integral equality, because functions that agree almost everywhere have the same Lebesgue integral. It does affect pointwise formulations, which ordinarily use the uncompleted product sigma-algebra before passing to equivalence classes modulo null sets.