Tonelli's and Fubini's Theorems

In the setting of Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be measure spaces with $\mu$ and $\nu$ $\sigma$-finite.

(i) Tonelli's theorem. Let $f\in\mathcal{M}^{+}(X\times Y,\mathcal{F}\otimes\mathcal{G})$. Then $f_{x}\in\mathcal{M}^{+}(Y,\mathcal{G})$ and $f^{y}\in\mathcal{M}^{+}(X,\mathcal{F})$ for all $x\in X$ and $y\in Y$, the functions $x\mapsto\int_{Y}f_{x}\,d\nu$ and $y\mapsto\int_{X}f^{y}\,d\mu$ lie in $\mathcal{M}^{+}(X,\mathcal{F})$ and $\mathcal{M}^{+}(Y,\mathcal{G})$, and

$$\int_{X\times Y}f\,d(\mu\otimes\nu)=\int_{X}\Bigl(\int_{Y}f_{x}\,d\nu\Bigr)d\mu(x)=\int_{Y}\Bigl(\int_{X}f^{y}\,d\mu\Bigr)d\nu(y).$$

(ii) Fubini's theorem. Let $f\in\mathcal{L}^{1}(\mu\otimes\nu)$. The set

$$N=\{x\in X:f_{x}\notin\mathcal{L}^{1}(\nu)\}$$

lies in $\mathcal{F}$ and $\mu(N)=0$; the function $I:X\to\mathbb{R}$ with $I(x)=\int_{Y}f_{x}\,d\nu$ for $x\in X\setminus N$ and $I(x)=0$ for $x\in N$ lies in $\mathcal{L}^{1}(\mu)$; and

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

(iii) Fubini's theorem in the other order. Let $f\in\mathcal{L}^{1}(\mu\otimes\nu)$. The set

$$M=\{y\in Y:f^{y}\notin\mathcal{L}^{1}(\mu)\}$$

lies in $\mathcal{G}$ and $\nu(M)=0$; the function $J:Y\to\mathbb{R}$ with $J(y)=\int_{X}f^{y}\,d\mu$ for $y\in Y\setminus M$ and $J(y)=0$ for $y\in M$ lies in $\mathcal{L}^{1}(\nu)$; and

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