Tonelli's theorem is a fundamental result in measure theory that justifies interchanging the order of integration for non-negative measurable functions defined on the product of two σ-finite measure spaces.[1]In precise terms, the theorem states that if (X, \mathcal{S}, \mu) and (Y, \mathcal{T}, \nu) are σ-finite measure spaces and f: X \times Y \to [0, \infty] is measurable with respect to the product σ-algebra \mathcal{S} \otimes \mathcal{T}, then the functions x \mapsto \int_Y f(x,y) \, d\nu(y) and y \mapsto \int_X f(x,y) \, d\mu(x) are measurable, and the following equalities hold:\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),where the integrals may be infinite.[2][3][1]This result extends naturally to countable products and serves as a key tool for computing multiple integrals by reducing them to iterated single integrals, particularly in applications involving Lebesgue measure on \mathbb{R}^n.[4] Unlike Fubini's theorem, which requires the function to be integrable (i.e., \int |f| \, d(\mu \times \nu) < \infty) and applies to signed functions, Tonelli's theorem relaxes the integrability condition by restricting to non-negative functions, avoiding issues with conditional convergence.[2][1]Tonelli's theorem was proved by the Italian mathematician Leonida Tonelli in 1909, building on earlier work by Henri Lebesgue in 1904 for bounded measurable functions and Guido Fubini in 1907 for integrable functions.[2][1] It plays a crucial role in the development of modern integration theory, enabling the rigorous justification of many calculations in analysis, probability, and partial differential equations that rely on product measures.[3]
Introduction
Statement
Tonelli's theorem provides a justification for interchanging the order of integration in the context of product measure spaces. Let (X, \mathcal{A}, \mu) and (Y, \mathcal{B}, \nu) be \sigma-finite measure spaces, and let f: X \times Y \to [0, \infty] be a function that is measurable with respect to the product \sigma-algebra \mathcal{A} \otimes \mathcal{B}. Then, the sections y \mapsto f(x, y) are \mathcal{B}-measurable for \mu-almost every x \in X, the sections x \mapsto f(x, y) are \mathcal{A}-measurable for \nu-almost every y \in Y, and the iterated integrals exist (possibly infinite) with\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).[3][5]The product measure \mu \times \nu on the product \sigma-algebra \mathcal{A} \otimes \mathcal{B} (generated by the measurable rectangles A \times B with A \in \mathcal{A} and B \in \mathcal{B}) is defined such that (\mu \times \nu)(A \times B) = \mu(A) \nu(B) for all such rectangles, and it extends uniquely to the entire \sigma-algebra under the \sigma-finiteness of \mu and \nu.[5][6] The \sigma-finiteness condition ensures that X and Y can be covered by countably many sets of finite measure, which is crucial for the existence and uniqueness of the product measure.[3]The non-negativity of f allows the integrals to be well-defined without requiring absolute integrability, distinguishing Tonelli's theorem from more restrictive results, and the equality holds even if one or both iterated integrals are infinite.[6][5]
Historical context
Tonelli's theorem is attributed to the Italian mathematician Leonida Tonelli (1885–1946), who proved it in 1909 amid his broader research on the calculus of variations, where multiple integrals frequently arise in problems involving functionals over multidimensional domains.[7] Tonelli's contribution addressed limitations in prior results by focusing on non-negative functions, enabling the reliable computation of iterated integrals without demanding finite total variation.[1]The theorem extends Guido Fubini's 1907 result on multiple integrals, which assumed absolute integrability to permit order interchange, by weakening this to non-negativity alone, thus broadening applicability in early measure-theoretic contexts.[8] Fubini's work had laid essential groundwork for product measures, but Tonelli's relaxation resolved cases where integrability was uncertain, marking a pivotal shift toward more flexible integration techniques.[1]Tonelli detailed his proof in the seminal paper "Sull'integrazione per parti," published in the Rendiconti dell'Accademia Nazionale dei Lincei (series 5), volume 18, pages 246–253.[8] This publication solidified the theorem's foundations within Italian mathematical circles, influencing subsequent developments in real analysis.[9]The theorem's introduction helped cement modern integration theory in the early 20th century, offering a robust method to interchange limits and integrals under minimal conditions, which proved instrumental in advancing measure theory and resolving longstanding issues in multidimensional calculus.[1] Its impact extended to variational problems, where Tonelli applied it to establish existence results for minimizers, contributing to the rigor of functional analysis.[7]
Background concepts
Lebesgue measure and integration
The Lebesgue outer measure on \mathbb{R}^n is defined for any set E \subset \mathbb{R}^n as \mu^*(E) = \inf \left\{ \sum_{i=1}^\infty \mu(R_i) : E \subset \bigcup_{i=1}^\infty R_i, \, R_i \in \mathcal{R}(\mathbb{R}^n) \right\}, where \mathcal{R}(\mathbb{R}^n) denotes the collection of n-dimensional rectangles and \mu(R) is the volume of the rectangle R.[10] This outer measure satisfies \mu^*(\emptyset) = 0, monotonicity (E \subset F implies \mu^*(E) \leq \mu^*(F)), and countable subadditivity (\mu^*\left(\bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu^*(E_i)).[10]A set A \subset \mathbb{R}^n is Lebesgue measurable if for every E \subset \mathbb{R}^n, \mu^*(E) = \mu^*(E \cap A) + \mu^*(E \cap A^c), where A^c is the complement of A.[10] The collection \mathcal{L}(\mathbb{R}^n) of all Lebesgue measurable sets forms a \sigma-algebra on \mathbb{R}^n, and the restriction of \mu^* to \mathcal{L}(\mathbb{R}^n) is a countably additive measure, known as the Lebesgue measure \mu.[10] This \sigma-algebra contains all open sets, closed sets, and Borel sets, and is complete, meaning any subset of a set of measure zero is measurable with measure zero.[10]A function f: \mathbb{R}^n \to \overline{\mathbb{R}} (where \overline{\mathbb{R}} is the extended reals) is Lebesgue measurable if the preimage f^{-1}(B) \in \mathcal{L}(\mathbb{R}^n) for every Borel set B \subset \mathbb{R}, or equivalently, if \{x : f(x) < a\} \in \mathcal{L}(\mathbb{R}^n) for every a \in \mathbb{R}.[11] Any non-negative measurable function f: \mathbb{R}^n \to [0, \infty] can be approximated pointwise by an increasing sequence of simple functions \{\phi_k\}, where each \phi_k(x) = \sum_{i=1}^{m_k} c_{k,i} \chi_{E_{k,i}}(x) with c_{k,i} \geq 0, E_{k,i} \in \mathcal{L}(\mathbb{R}^n), and \phi_k \uparrow f.[11] If f is bounded on a set of finite measure, the convergence is uniform on that set.[11]The Lebesgue integral of a non-negative measurable function f is defined as \int_{\mathbb{R}^n} f \, d\mu = \sup \left\{ \int_{\mathbb{R}^n} \phi \, d\mu : 0 \leq \phi \leq f, \, \phi \text{ simple} \right\}, where the integral of a simple function \phi = \sum c_i \chi_{E_i} is \int \phi \, d\mu = \sum c_i \mu(E_i).[11] For signed measurable functions f = f^+ - f^- with f^+, f^- \geq 0 and at least one of \int f^+ or \int f^- finite, the integral extends as \int f \, d\mu = \int f^+ \, d\mu - \int f^- \, d\mu.[11] This construction ensures linearity and preserves measurability under pointwise limits and algebraic operations.[11]A measure space (\mathbb{R}^n, \mathcal{L}(\mathbb{R}^n), \mu) is \sigma-finite if \mathbb{R}^n = \bigcup_{k=1}^\infty A_k with each A_k \in \mathcal{L}(\mathbb{R}^n) and \mu(A_k) < \infty, as holds for Lebesgue measure since \mathbb{R}^n = \bigcup_{k=1}^\infty B(0,k) where B(0,k) is the ball of radius k with finite volume.[12] \sigma-finiteness is essential for countable additivity to support limiting processes in integration and ensures the measure behaves well on infinite spaces like \mathbb{R}^n, facilitating theorems that rely on finite approximations.[12]
Product measure spaces
In measure theory, the product of two measurable spaces (X, \mathcal{A}, \mu) and (Y, \mathcal{B}, \nu) is the Cartesian product space X \times Y equipped with a suitable σ-algebra and measure that extend the structures from the individual spaces.[2][13]The product σ-algebra \mathcal{A} \otimes \mathcal{B} on X \times Y is the smallest σ-algebra containing all measurable rectangles of the form A \times B, where A \in \mathcal{A} and B \in \mathcal{B}.[2][14] This σ-algebra is generated by the collection of such rectangles and includes all finite disjoint unions of them, ensuring that projections from X \times Y to X and Y are measurable.[2]Given σ-finite measures \mu and \nu, the product measure \mu \times \nu on \mathcal{A} \otimes \mathcal{B} is constructed by first defining it on the measurable rectangles as \mu \times \nu(A \times B) = \mu(A) \nu(B).[2][13] This definition extends to finite disjoint unions of rectangles via additivity, forming a premeasure on the algebra generated by the rectangles, which is then uniquely extended to a measure on the full product σ-algebra using Carathéodory's extension theorem.[14]The uniqueness of this extension holds when \mu and \nu are σ-finite, meaning each space can be covered by a countable union of sets of finite measure.[2][13] This uniqueness can be established using the monotone class theorem, which shows that any measure agreeing on the algebra of finite disjoint unions of rectangles must coincide on the generated σ-algebra.[2][14]For a set E \in \mathcal{A} \otimes \mathcal{B}, the sections are defined as E_x = \{ y \in Y \mid (x, y) \in E \} for fixed x \in X and E^y = \{ x \in X \mid (x, y) \in E \} for fixed y \in Y; under the product measure, these sections belong to \mathcal{B} and \mathcal{A}, respectively.[2][14] Similarly, for a measurable function f: X \times Y \to \mathbb{R}, the sections f_x(y) = f(x, y) and f^y(x) = f(x, y) are measurable functions on Y and X, respectively.[2][14]
The theorem
Formal statement
Tonelli's theorem provides a precise condition under which the double integral of a non-negative measurable function over a product measure space equals the iterated integrals in either order.[15]Let (X, \mathcal{A}, \mu) and (Y, \mathcal{B}, \nu) be \sigma-finite measure spaces, and let f: X \times Y \to [0, \infty] be (\mathcal{A} \otimes \mathcal{B})-measurable. Then the functions x \mapsto \int_Y f(x,y) \, d\nu(y) and y \mapsto \int_X f(x,y) \, d\mu(x) are measurable (with values in [0, \infty]), and\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).[15][16]The equality of the iterated integrals holds even if one or both are infinite.[15]Moreover, if at least one of the iterated integrals is finite, then both are finite and f is integrable with respect to the product measure \mu \times \nu.[17]
Key assumptions
Tonelli's theorem requires the integrand to be a non-negative measurable function, which enables the use of the monotone convergence theorem during the proof. This assumption circumvents the need for absolute convergence that is necessary for signed functions in Fubini's theorem, allowing the theorem to apply to a broader class of functions where the integral may be infinite. By approximating the non-negative function with an increasing sequence of simple functions, the monotone convergence theorem justifies passing the limit inside the integrals, ensuring the equality of the iterated and double integrals holds reliably.[18]The measurability of the function with respect to the product σ-algebra is essential to guarantee that the sections of the function—fixed in one variable—are measurable with respect to the σ-algebra of the other space. This ensures that the iterated integrals are well-defined over the respective measure spaces, as non-measurable sections would prevent the computation of integrals or lead to undefined expressions. Lebesgue measurability, as defined in the background, underpins this requirement by providing the framework for such sections to inherit measurability properties.[19]The σ-finiteness of both measure spaces is a fundamental hypothesis that ensures the existence and uniqueness of the product measure, which must itself be σ-finite to support the theorem's constructions. Without σ-finiteness, the product measure may not be well-behaved, leading to failures where the iterated integrals disagree or one evaluates to a finite value while the double integral is infinite; a classic counterexample involves the Lebesgue measure on [0,1] paired with the counting measure on the uncountable set [0,1], where the product measure assigns infinity to certain sets despite finite iterated integrals over characteristic functions of the diagonal. This assumption allows the proof to proceed by partitioning the spaces into countably many finite-measure sets, facilitating the application of monotone convergence on each piece.[20]Although Tonelli's theorem is typically stated for σ-finite measures without requiring completeness of the spaces, the result holds in this setting, and extensions to incomplete measures are possible through completion procedures that preserve the key equalities.[19]
Proof
Outline of the proof
The proof of Tonelli's theorem relies on a structured reduction from simple cases to the general one, exploiting the countable additivity of the product measure and monotonicity principles for non-negative functions.[3]It begins by verifying the theorem for indicator functions of measurable rectangles in the product space, where the double integral equals the product of the measures of the individual sets, and the iterated integrals follow directly from this by additivity.[3]Next, the result extends to non-negative simple functions—finite sums of non-negative multiples of such indicators—via the linearity of integration over the product measure.[3]For a general non-negative measurable function, an increasing sequence of non-negative simple functions approximating it pointwise is used, and the monotone convergence theorem ensures the integrals converge to the desired equality between the double integral and the iterated integrals.[3]Throughout, the measurability of the section functions (the inner integrals as functions on each factor space) is established using properties of measurable sections in product σ-algebras, confirming the iterated integrals are well-defined.[3]This approach highlights how countable additivity handles the rectangle case, while monotonicity bridges to arbitrary non-negative measurables without requiring absolute integrability.[3]
Detailed construction
To establish Tonelli's theorem, the proof begins with a key lemma ensuring the measurability of sections, which is essential for defining the iterated integrals.[2]Lemma (Measurability of Sections). Let (X, \mathcal{M}, \mu) and (Y, \mathcal{N}, \nu) be \sigma-finite measure spaces, and let f: X \times Y \to [0, \infty] be (\mathcal{M} \otimes \mathcal{N})-measurable. Then, the section functions f_x: Y \to [0, \infty], defined by f_x(y) = f(x, y), are \mathcal{N}-measurable for every x \in X. Similarly, the sections f^y: X \to [0, \infty], defined by f^y(x) = f(x, y), are \mathcal{M}-measurable for every y \in Y.[2][21]This lemma follows from the measurability of sections for measurable sets in the product \sigma-algebra and extension to functions via standard arguments for non-negative measurables. Specifically, for a measurable set E \subset X \times Y, the sections E_x = \{y \in Y : (x, y) \in E\} are in \mathcal{N} for every x \in X, and E^y = \{x \in X : (x, y) \in E\} are in \mathcal{M} for every y \in Y; the same holds for functions by applying this to level sets \{(x, y) : f(x, y) > t\}.[2]With the sections measurable everywhere, the iterated integrals \int_X \left( \int_Y f_x \, d\nu \right) d\mu and \int_Y \left( \int_X f^y \, d\mu \right) d\nu are well-defined (finite or infinite). The proof now proceeds by first verifying the theorem for non-negative simple functions and then extending via monotone approximation.[21]Consider a non-negative simple function s = \sum_{k=1}^n c_k \mathbf{1}_{A_k \times B_k}, where c_k \geq 0, each A_k \in \mathcal{M}, and each B_k \in \mathcal{N}. The section s_x(y) = \sum_{k=1}^n c_k \mathbf{1}_{A_k}(x) \mathbf{1}_{B_k}(y), so \int_Y s_x \, d\nu = \sum_{k=1}^n c_k \mathbf{1}_{A_k}(x) \nu(B_k). Integrating over X yields\int_X \left( \int_Y s_x \, d\nu \right) d\mu = \sum_{k=1}^n c_k \nu(B_k) \int_X \mathbf{1}_{A_k} \, d\mu = \sum_{k=1}^n c_k \mu(A_k) \nu(B_k).By symmetry, the other iterated integral \int_Y \left( \int_X s^y \, d\mu \right) d\nu equals the same expression. For the double integral, recall that the product measure \mu \times \nu on rectangles satisfies (\mu \times \nu)(A_k \times B_k) = \mu(A_k) \nu(B_k), and by linearity,\int_{X \times Y} s \, d(\mu \times \nu) = \sum_{k=1}^n c_k (\mu \times \nu)(A_k \times B_k) = \sum_{k=1}^n c_k \mu(A_k) \nu(B_k).Thus, the iterated and double integrals coincide for simple functions.[2][21]For a general non-negative measurable f, there exists a sequence of non-negative simple functions \{s_n\}_{n=1}^\infty such that s_n \uparrow f pointwise on X \times Y. By the monotone convergence theorem applied to the sections,\int_Y f_x \, d\nu = \lim_{n \to \infty} \int_Y (s_n)_x \, d\nu \quad \text{for every } x \in X,and thus\int_X \left( \int_Y f_x \, d\nu \right) d\mu = \lim_{n \to \infty} \int_X \left( \int_Y (s_n)_x \, d\nu \right) d\mu = \lim_{n \to \infty} \int_{X \times Y} s_n \, d(\mu \times \nu),with the final limit justified by monotone convergence on the product space. The same holds for the other iterated integral. Since the double integral is defined as the supremum over integrals of simple functions below f, it equals \lim_{n \to \infty} \int_{X \times Y} s_n \, d(\mu \times \nu). Therefore, all three integrals are equal.[2][21][22]
Relation to other theorems
Comparison with Fubini's theorem
Fubini's theorem addresses the interchange of integration order for signed functions on product measure spaces, requiring that the function be absolutely integrable, meaning the integral of its absolute value is finite.[22] This absolute integrability condition ensures that both iterated integrals exist, are finite, and equal the double integral over the product space.[22]In contrast, Tonelli's theorem applies specifically to non-negative measurable functions and does not require absolute integrability; it allows the iterated integrals to be infinite as long as they are equal.[22] The key difference lies in the hypotheses: Tonelli's weaker conditions of non-negativity and measurability suffice for order interchange, even when integrals diverge, whereas Fubini's stricter L¹ requirement handles signed functions but fails without it.[22]Tonelli's theorem implies Fubini's for non-negative functions, and extends to signed functions via decomposition: any measurable function f can be written as f = f⁺ - f⁻, where f⁺ and f⁻ are non-negative; applying Tonelli to each part shows that if ∫|f| < ∞, then the integrals of f⁺ and f⁻ are finite and Fubini's conditions hold.[22]Without absolute integrability, Fubini's theorem can fail dramatically, as iterated integrals may exist but differ in value due to conditional convergence, resembling alternating series. For example, consider f(x, y) = (x² - y²)/(x² + y²)² on [0,1] × [0,1] with Lebesgue measure; the iterated integral ∫[0,1] (∫[0,1] f(x,y) dy) dx = π/4, while ∫[0,1] (∫[0,1] f(x,y) dx) dy = -π/4, yet ∫∫ |f| = ∞, violating Fubini's hypothesis.[23] In this case, Tonelli's theorem applies to |f|, confirming both iterated integrals of |f| equal ∞, highlighting its robustness for non-negative cases where Fubini does not.[22]
Generalizations
Tonelli's theorem extends naturally to integrals over n-fold product measure spaces for n \geq 2. By iteratively applying the two-variable case, the theorem asserts that for a non-negative measurable function f on the product space X_1 \times \cdots \times X_n equipped with σ-finite measures \mu_1, \dots, \mu_n, the multiple integral equals any iterated integral in arbitrary order, provided the integrals exist (possibly infinite). This iterative extension relies on the σ-finiteness of each measure to ensure the product measure is well-defined and the sections remain measurable.[24]For non-σ-finite measures, the theorem does not hold in general, but partial generalizations exist under weaker conditions such as semifiniteness. Specifically, if one measure is σ-finite and the other is semifinite, Tonelli's theorem holds for non-negative measurable functions if and only if the induced product outer measure is semifinite, meaning that any set of infinite measure contains a subset of finite positive measure. Semifiniteness can replace σ-finiteness in one space if the sections of measurable sets are measurable with respect to the other space's σ-algebra. These results connect the validity of the theorem to properties of the product measure construction.[25]In abstract σ-finite measure spaces, Tonelli's theorem applies directly, enabling the interchange of integrals for non-negative functions without additional restrictions beyond measurability. In probability theory, where measures are typically probability measures (hence σ-finite), the theorem underpins the computation of expectations for non-negative random variables over product spaces; for instance, it justifies \mathbb{E}[X Y] = \mathbb{E}[X] \mathbb{E}[Y] for independent non-negative random variables X and Y by viewing the expectation as an integral over the product probability space. This framework is foundational for handling iterated expectations in stochastic processes.[26]Connections to ergodic theory arise through applications of Tonelli's theorem in proving ergodic decompositions and averaging results, where non-negative functions on product spaces model invariant measures and orbit averages, though such uses remain within the classical σ-finite framework.[27]
Examples and applications
Basic examples
To illustrate Tonelli's theorem, consider the constant non-negative function f(x,y) = 1 defined on the unit square [0,1] \times [0,1] equipped with the product Lebesgue measure. The double integral is \iint_{[0,1]^2} 1 \, d\lambda_2 = 1, where \lambda_2 denotes the two-dimensional Lebesgue measure. The iterated integrals are \int_0^1 \left( \int_0^1 1 \, dy \right) dx = \int_0^1 1 \, dx = 1 and \int_0^1 \left( \int_0^1 1 \, dx \right) dy = 1, confirming equality as predicted by the theorem for non-negative measurable functions.[22]A more interesting case arises with the non-negative function f(x,y) = \frac{1}{(x+y)^2} on (0,\infty) \times (0,\infty) under the product Lebesgue measure, where the integrals are improper. For fixed x > 0, the inner integral is \int_0^\infty \frac{1}{(x+y)^2} \, dy = \left[ -\frac{1}{x+y} \right]_0^\infty = \frac{1}{x}, so the iterated integral becomes \int_0^\infty \frac{1}{x} \, dx = \infty. By symmetry, the reverse order yields the same infinite value, and the double integral is also \infty, demonstrating that Tonelli's theorem extends to cases where all integrals diverge equally.Another straightforward application involves the indicator function of the diagonal set, f(x,y) = \chi_{\{ (x,y) : x = y \}} (x,y) on [0,1] \times [0,1] with product Lebesgue measure. The diagonal subset has two-dimensional Lebesgue measure zero, so the double integral is \iint_{[0,1]^2} f \, d\lambda_2 = 0. For the iterated integrals, fixing x \in [0,1], the inner integral \int_0^1 \chi_{\{y = x\}} (x,y) \, dy = 0 since it measures a single point, yielding \int_0^1 0 \, dx = 0; the reverse order is analogous. Thus, equality holds at zero, underscoring the theorem's validity for non-negative functions supported on negligible sets.[28]The non-negativity condition in Tonelli's theorem is essential, as its absence can lead to discrepancies. For instance, consider the signed function f(x,y) = \frac{xy(x^2 - y^2)}{(x^2 + y^2)^3} for (x,y) \neq (0,0) and f(0,0) = 0 on the rectangle [0,2] \times [0,1] with Lebesgue measure. The iterated integral first with respect to y is \int_0^2 \left( \int_0^1 f(x,y) \, dy \right) dx = \int_0^2 \frac{x}{(x^2 + 1)^2} \, dx = \frac{1}{5}, while first with respect to x gives \int_0^1 \left( \int_0^2 f(x,y) \, dx \right) dy = \int_0^1 \frac{-2y}{(4 + y^2)^2} \, dy = -\frac{1}{20}. The values differ because f changes sign and is not absolutely integrable over the domain (it is unbounded near the origin), so Tonelli's theorem does not apply.[29]
Applications in analysis
In probability theory, Tonelli's theorem facilitates the interchange of expectation and conditioning for non-negative random variables, ensuring that the law of total expectation holds without requiring absolute integrability. Specifically, for a non-negative random variable X and a sub-\sigma-algebra \mathcal{G}, the conditional expectation satisfies \mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid \mathcal{G}]], as the iterated integral over the product probability space aligns with the double integral via Tonelli's application to the non-negative indicator functions defining the expectations.[30]In partial differential equations (PDEs), Tonelli's theorem justifies the evaluation of iterated integrals involving Green's functions for equations like the heat equation, particularly when verifying continuity or boundedness of solutions through double integrals over space and time. For instance, in analyzing the stochastic heat equation, it allows interchanging the order of integration in expressions like \int_0^t \int_{-\infty}^\infty |\Gamma(t-s; y) - \Gamma(t-s; x' - x - y)|^2 \, dy \, ds, where \Gamma denotes the heat kernel, confirming the L^2 norm's finiteness for the difference of Green's functions.[31]In harmonic analysis, Tonelli's theorem supports computations of norms for functions on product spaces, such as in Fourier transforms over multiple variables, by permitting the interchange of integrals in Littlewood-Paley decompositions or approximations by Fourier kernels. This is essential for bounding L^p norms, as seen in estimates like \|K_t * \phi - \phi\|_1, where swapping integration orders via Tonelli ensures the convolution's convergence and norm control in product measure settings.[32]Tonelli's theorem extends to stochastic calculus as a precursor to Itô integrals, enabling stochastic Fubini-Tonelli versions that justify interchanging expectations, time integrals, and stochastic integrals for non-negative integrands in processes like semimartingales. In the Itô-Henstock integral framework, it underpins theorems allowing such swaps, crucial for defining and computing integrals in non-standard stochastic settings without relying on L^2 assumptions.[33]In modern data science and machine learning, Tonelli's theorem (often via Fubini-Tonelli) aids in computing expected values for loss functions or spectral densities in neural networks, such as interchanging integrals in the moments method for Gram matrices M = \frac{1}{m} Y^T Y with Y = f(WX). This supports asymptotic analysis of eigenvalue distributions and generalization bounds by reordering bounded multi-dimensional integrals over parameters like \lambda and z.[34]A key example arises in geometric measure theory, where Tonelli's theorem computes the volume (Lebesgue measure) of sets via integration over slices, equating the measure of a product set to iterated integrals of section measures. For a measurable set E \subset \mathbb{R}^n \times \mathbb{R}^m, this yields \mathcal{L}^{n+m}(E) = \int_{\mathbb{R}^n} \mathcal{L}^m(E_x) \, d\mathcal{L}^n(x) = \int_{\mathbb{R}^m} \mathcal{L}^n(E^y) \, d\mathcal{L}^m(y), foundational for analyzing rectifiability and Hausdorff measures of higher-dimensional objects.[3]