Proof of Tonelli's and Fubini's Theorems

Throughout, the section at $x\in X$ of a function on $X\times Y$ is its composite with the insertion $y\mapsto(x,y)$ (Precomposition Commutes with Pointwise Operations, and Sections of Functions Are Precompositions (iv)), and composites commute with the pointwise operations and send indicators to indicators of preimages, here of sections (Precomposition Commutes with Pointwise Operations, and Sections of Functions Are Precompositions (ii), Precomposition Commutes with Pointwise Operations, and Sections of Functions Are Precompositions (i), Rectangles, Sections and the Swap: Set Identities (iii)). So for functions $f,g$ on $X\times Y$, $c\in[0,\infty]$, $E\subseteq X\times Y$ and $x\in X$,

$$(f+g)_{x}=f_{x}+g_{x},\qquad(cf)_{x}=cf_{x},\qquad(\mathbf{1}_{E})_{x}=\mathbf{1}_{E_{x}},\qquad|f|_{x}=|f_{x}|,\qquad(f^{\pm})_{x}=(f_{x})^{\pm},$$

and if $(f_{m})$ is a sequence of functions $X\times Y\to[0,\infty]$ and $h(p)=\sup_{m}f_{m}(p)$ for every $p\in X\times Y$, then $h_{x}(y)=\sup_{m}(f_{m})_{x}(y)$ for every $y\in Y$; likewise at $y\in Y$. Sections of measurable functions are measurable (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (v)): for $f\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$, $f_{x}\in\mathcal{M}^{+}(Y,\mathcal{G})$ and $f^{y}\in\mathcal{M}^{+}(X,\mathcal{F})$, and for a measurable $f:X\times Y\to\mathbb{R}$, $f_{x}:Y\to\mathbb{R}$ and $f^{y}:X\to\mathbb{R}$ are measurable.

Proof of (i). We prove that $x\mapsto\int_{Y}f_{x}\,d\nu$ lies in $\mathcal{M}^{+}(X,\mathcal{F})$ and that the first two integrals in the display agree. The statements about $y\mapsto\int_{X}f^{y}\,d\mu$ follow by the same argument with sections at $y$ in place of sections at $x$, using the second formula of Existence and Uniqueness of the Product Measure (iii) in Step 1.

For $f\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$ write $\Phi_{f}(x)=\int_{Y}f_{x}\,d\nu$ for $x\in X$, and let $\mathcal{H}$ be the set of those $f\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$ for which

$$\Phi_{f}\in\mathcal{M}^{+}(X,\mathcal{F})\qquad\text{and}\qquad\int_{X}\Phi_{f}\,d\mu=\int_{X\times Y}f\,d(\mu\otimes\nu).$$

We show $\mathcal{H}=\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$.

Step 1: indicators. Let $E\in\mathcal{F}\otimes\mathcal{G}$. Then $\mathbf{1}_{E}\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$ and, as $E_{x}\in\mathcal{G}$ (The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap (iv)), $\Phi_{\mathbf{1}_{E}}(x)=\int_{Y}\mathbf{1}_{E_{x}}\,d\nu=\nu(E_{x})$ (Basic Properties of the Integral of Nonnegative Measurable Maps (ii), on both spaces). By Existence and Uniqueness of the Product Measure (iii), $\Phi_{\mathbf{1}_{E}}\in\mathcal{M}^{+}(X,\mathcal{F})$ and

$$\int_{X}\Phi_{\mathbf{1}_{E}}\,d\mu=\int_{X}\nu(E_{x})\,d\mu(x)=(\mu\otimes\nu)(E)=\int_{X\times Y}\mathbf{1}_{E}\,d(\mu\otimes\nu).$$

So $\mathbf{1}_{E}\in\mathcal{H}$.

Step 2: multiples. Let $f\in\mathcal{H}$ and let $c\ge0$ be real. Then $(cf)_{x}=cf_{x}$, and Basic Properties of the Integral of Nonnegative Measurable Maps (iv) on $Y$, on $X$ and on $X\times Y$ gives $\Phi_{cf}=c\,\Phi_{f}\in\mathcal{M}^{+}(X,\mathcal{F})$ and

$$\int_{X}\Phi_{cf}\,d\mu=c\int_{X}\Phi_{f}\,d\mu=c\int_{X\times Y}f\,d(\mu\otimes\nu)=\int_{X\times Y}cf\,d(\mu\otimes\nu).$$

So $cf\in\mathcal{H}$.

Step 3: nonnegative simple functions. Let $s\in\mathcal{S}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$, let $C=s[X\times Y]$, a finite set of nonnegative reals, and let $E_{c}=s^{-1}(\{c\})\in\mathcal{F}\otimes\mathcal{G}$ for $c\in C$ (Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv)). Then

$$s=\sum_{c\in C}c\,\mathbf{1}_{E_{c}},\qquad s_{x}=\sum_{c\in C}c\,\mathbf{1}_{(E_{c})_{x}}\quad(x\in X)$$

pointwise (at $p\in X\times Y$ the only term that can be nonzero is the one with $c=s(p)$, Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iii)), and each $c\,\mathbf{1}_{E_{c}}$ lies in $\mathcal{H}$ by Steps 1 and 2. The integral of a finite sum in $\mathcal{M}^{+}$ is the sum of the integrals (Integrals of Finite Sums of Nonnegative Functions, and Constant Multiples of a Measure (i)). Applied on $Y$, this gives $\Phi_{s}=\sum_{c\in C}\Phi_{c\mathbf{1}_{E_{c}}}$; applied on $X$, $\Phi_{s}\in\mathcal{M}^{+}(X,\mathcal{F})$; and applied on $X$ and on $X\times Y$,

$$\int_{X}\Phi_{s}\,d\mu=\sum_{c\in C}\int_{X}\Phi_{c\mathbf{1}_{E_{c}}}\,d\mu=\sum_{c\in C}\int_{X\times Y}c\,\mathbf{1}_{E_{c}}\,d(\mu\otimes\nu)=\int_{X\times Y}s\,d(\mu\otimes\nu).$$

So $s\in\mathcal{H}$.

Step 4: all of $\mathcal{M}^{+}$. Let $f\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$, and let $s_{m}=s_{m}(f)$ be its dyadic approximations: $s_{m}\in\mathcal{S}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$, $s_{m}\le s_{m+1}$ and $f=\sup_{m}s_{m}$ pointwise (Approximation of Measurable Functions by Dyadic Simple Functions (i)). By Step 3, $s_{m}\in\mathcal{H}$ for every $m$. For $x\in X$ the sections $(s_{m})_{x}$ increase to $f_{x}$, so the monotone convergence theorem on $(Y,\mathcal{G},\nu)$ (Monotone Convergence Theorem (ii)) gives

$$\Phi_{f}(x)=\sup_{m}\Phi_{s_{m}}(x).$$

The functions $\Phi_{s_{m}}\in\mathcal{M}^{+}(X,\mathcal{F})$ increase with $m$ (Basic Properties of the Integral of Nonnegative Measurable Maps (iii)), so $\Phi_{f}\in\mathcal{M}^{+}(X,\mathcal{F})$ (Monotone Convergence Theorem (i)), and monotone convergence on $(X,\mathcal{F},\mu)$ and on $(X\times Y,\mathcal{F}\otimes\mathcal{G},\mu\otimes\nu)$ (Monotone Convergence Theorem (ii)) gives

$$\int_{X}\Phi_{f}\,d\mu=\sup_{m}\int_{X}\Phi_{s_{m}}\,d\mu=\sup_{m}\int_{X\times Y}s_{m}\,d(\mu\otimes\nu)=\int_{X\times Y}f\,d(\mu\otimes\nu).$$

So $f\in\mathcal{H}$, which proves (i).

Proof of (ii). Let $f\in\mathcal{L}^{1}(\mu\otimes\nu)$. Then $|f|,f^{+},f^{-}\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$ and

$$\int_{X\times Y}|f|\,d(\mu\otimes\nu)=\int_{X\times Y}f^{+}\,d(\mu\otimes\nu)+\int_{X\times Y}f^{-}\,d(\mu\otimes\nu)<\infty$$

(Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv)).

The exceptional set. Let $G(x)=\int_{Y}|f_{x}|\,d\nu$. By (i) applied to $|f|$, $G\in\mathcal{M}^{+}(X,\mathcal{F})$ and $\int_{X}G\,d\mu=\int_{X\times Y}|f|\,d(\mu\otimes\nu)<\infty$. By the integrability criterion on $(Y,\mathcal{G},\nu)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv)), $f_{x}\in\mathcal{L}^{1}(\nu)$ if and only if $G(x)<\infty$, so

$$N=\{x\in X:G(x)=\infty\},$$

which lies in $\mathcal{F}$ and has $\mu(N)=0$ by The Lebesgue Integral and Null Sets (v).

The function $I$. Let $P(x)=\int_{Y}(f_{x})^{+}\,d\nu$ and $Q(x)=\int_{Y}(f_{x})^{-}\,d\nu$. By (i) applied to $f^{+}$ and $f^{-}$, $P,Q\in\mathcal{M}^{+}(X,\mathcal{F})$ with

$$\int_{X}P\,d\mu=\int_{X\times Y}f^{+}\,d(\mu\otimes\nu)<\infty,\qquad\int_{X}Q\,d\mu=\int_{X\times Y}f^{-}\,d(\mu\otimes\nu)<\infty.$$

Since $G=P+Q$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv) on $Y$), $P(x)$ and $Q(x)$ are real for $x\notin N$. Let $\tilde P=P\,\mathbf{1}_{X\setminus N}$ and $\tilde Q=Q\,\mathbf{1}_{X\setminus N}$; they are real-valued and, as $\mathbf{1}_{X\setminus N}\in\mathcal{M}^{+}(X,\mathcal{F})$ (Basic Properties of the Integral of Nonnegative Measurable Maps (ii)), lie in $\mathcal{M}^{+}(X,\mathcal{F})$ (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (iii)), and so are measurable as real functions (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)). As $\tilde P=P$ outside the null set $N$, $\int_{X}\tilde P\,d\mu=\int_{X}P\,d\mu<\infty$ (The Lebesgue Integral and Null Sets (ii)), so $\tilde P\in\mathcal{L}^{1}(\mu)$ with integral $\int_{X\times Y}f^{+}\,d(\mu\otimes\nu)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)); likewise $\tilde Q\in\mathcal{L}^{1}(\mu)$ with integral $\int_{X\times Y}f^{-}\,d(\mu\otimes\nu)$.

For $x\notin N$, the definition of the integral of $f_{x}\in\mathcal{L}^{1}(\nu)$ (Integrable Functions and Their Lebesgue Integral (iii)) gives $I(x)=P(x)-Q(x)=\tilde P(x)-\tilde Q(x)$, and for $x\in N$, $I(x)=0=\tilde P(x)-\tilde Q(x)$. Hence $I=\tilde P-\tilde Q\in\mathcal{L}^{1}(\mu)$ and, by linearity (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii)) and the definition of the integral of $f$ (Integrable Functions and Their Lebesgue Integral (iii)),

$$\int_{X}I\,d\mu=\int_{X}\tilde P\,d\mu-\int_{X}\tilde Q\,d\mu=\int_{X\times Y}f^{+}\,d(\mu\otimes\nu)-\int_{X\times Y}f^{-}\,d(\mu\otimes\nu)=\int_{X\times Y}f\,d(\mu\otimes\nu).$$

Proof of (iii). This is the argument for (ii) with sections at $y$ in place of sections at $x$, using the second equality of (i) in place of the first.