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I am looking for a direct proof of the "Fubini" theorem for the Darboux Integral.

The Theorem: Let $I_1\subseteq \mathbb{R}^n$, $I_2\subseteq \mathbb{R}^m$ be boxes and $f:I_1\times I_2\to \mathbb{R}$ be integrable. Then the iterated integrals \begin{equation}\int_{I_1}\left(\int_{I_2}f(x,y)\;dx\right)\;dy\text{ and }\int_{I_2}\left(\int_{I_1}f(x,y)\;dy\right)\;dx\end{equation} exist and \begin{equation}\int_{I_2\times I_2}f=\int_{I_1}\left(\int_{I_2}f(x,y)\;dx\right)\;dy=\int_{I_2}\left(\int_{I_1}f(x,y)\;dy\right)\;dx\end{equation}

A ($n$-th dimensional) box $I$ is a set $I= \left\{(x_1,...,x_n)\in \mathbb{R}^n:a_i\le x_i\le b_i,\ i=1,...,n\right\}$. The integral of a bounded function $f:I\to \mathbb{R}$ is defined as follows:

If $\mathcal{P}=\left\{ \mathbf{x}\in \mathbb{R}^n :c_{i-1,j}\le x_j\le c_{i,j}\ , j=1,...,n, i=1,...,k \right\}$ is a partition of $I$ with subpartitions $\mathcal{P}_i=\left\{\mathbf{x}\in \mathbb{R}^n :c_{i-1,j}\le x_j\le c_{i,j}\ , j=1,...,n \right\}$ we define the upper and lower Riemann sums of $f$ as \begin{equation} U_{f,\mathcal{P}}:=\sum\limits_{i=1}^k\sup_{\mathbf{x}\in \mathcal{P}_i}f(\mathbf{x})vol(\mathcal{P}_i) \text{ and } L_{f,\mathcal{P}}:=\sum\limits_{i=1}^k\inf_{\mathbf{x}\in \mathcal{P}_i}f(\mathbf{x})vol(\mathcal{P}_i) \end{equation} where $vol(\mathcal{P}_i)=\prod_{j=1}^{n}(c_{i,j}-c_{i-1,j})$. If the numbers \begin{equation}\int\limits_{I}^{*}f:=\inf_{\mathcal{P}}U_{f,\mathcal{P}} \text{ and } \int\limits_{*I}f:=\sup_{\mathcal{Q}}L_{f,\mathcal{Q}}\end{equation} are equal we say that $f$ is Riemann Integrable and denote their common value with the symbol $\int\limits_{I}f$.

As I already mentioned I am looking for a somewhat direct proof from this definition. Other proofs utilising the definition with step functions can be seen here: http://www.tau.ac.il/~tsirel/Courses/Analysis3/lect9.pdf, http://www.cmc.edu/math/publications/aksoy/Mixed_Partials.pdf, http://www.owlnet.rice.edu/~fjones/chap9.pdf

Based on http://math.berkeley.edu/~wodzicki/H104.F10/Integral.pdf pg 19 here is what I have done: Let $\mathcal{P}$ be a partition of $I_1\times I_2\subseteq \mathbb{R}^{n+m}$, \begin{equation}\mathcal{P}=\left\{ \mathbf{z}\in \mathbb{R}^{n+m}:c_{i-1,j}\le x_j\le c_{i,j}\text{ and }\notag\\c_{i-1,j^{\prime}}\le y_{j^{\prime}-n}\le c_{i,j^{\prime}}\ , j=1,...,n, j^{\prime}=n+1,...,n+m, i=1,...,k \right\}\end{equation} where $\mathbf{z}=(x_1,...,x_n,y_1,...,y_m)$. Consider the partitions $\mathcal{P}_1$, $\mathcal{P}_2$ of $I_1$ and $I_2$ respectively, \begin{gather}\mathcal{P}_1=\left\{ \mathbf{x}\in \mathbb{R}^{n}:c_{i-1,j}\le x_j\le c_{i,j}\ , j=1,...,n, i=1,...,k \right\}\text{ and }\notag\\ \mathcal{P}_2=\left\{ \mathbf{y}\in \mathbb{R}^{m}:c_{i-1,j^{\prime}}\le y_{j^{\prime}-n}\le c_{i,j^{\prime}}\ j^{\prime}=n+1,...,n+m, i=1,...,k \right\}\end{gather} Obviously $\mathcal{P}_{1i}\times\mathcal{P}_{2i}= \mathcal{P}_i$ and $\mathcal{P}_{1}\times\mathcal{P}_{2}= \mathcal{P}$. Then, for $i=1,...,k$ \begin{equation}\inf_{(x,y)\in \mathcal{P}_i}f(x,y)=\inf_{x\in \mathcal{P}_{1i}}\left(\inf_{y\in \mathcal{P}_{2i}}f(x,y)\right)\end{equation} Indeed, for arbitrary $\epsilon>0$, \begin{gather}\exists x\in \mathcal{P}_{1i}:\inf_{x\in \mathcal{P}_{1i}}\left(\inf_{y\in \mathcal{P}_{2i}}f(x,y)\right)+\frac{\epsilon}{2}>\inf_{y\in \mathcal{P}_{2i}}f(x,y)\text{ and } \exists y\in \mathcal{P}_{2i}:\inf_{y\in \mathcal{P}_{2i}}f(x,y)+\frac{\epsilon}{2}>f(x,y)\Rightarrow \notag\\ \exists (x,y)\in \mathcal{P}{i}:\inf_{x\in \mathcal{P}_{1i}}\left(\inf_{y\in \mathcal{P}_{2i}}f(x,y)\right)+\epsilon>f(x,y) \end{gather} Therefore, \begin{equation} L_{f,\mathcal{P}}=\sum\limits_{i=1}^k\inf_{(x,y)\in \mathcal{P}_i}f(x,y)vol(\mathcal{P}_i)=\sum\limits_{i=1}^k\inf_{x\in \mathcal{P}_{1i}}\left(\inf_{y\in \mathcal{P}_{2i}}f(x,y)\right)vol(\mathcal{P}_{1i})vol(\mathcal{P}_{2i}) \end{equation} How do I proceed from there?

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    I have a *slight* correction. What you have there is technically the [Darboux integral](https://en.wikipedia.org/wiki/Darboux_integral). The definition of the Riemann integral uses tagged partitions and is equivalent to the Darboux integral.2012-10-13
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    It is the Darboux definition of the Riemann Integral. Should I write that in the title to make things clearer?2012-10-13
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    @Nameless : Riemann's integral was defined by Riemann, not by Darboux. If Darboux defined something it is Darboux's integral, whether they are equivalent or not in whatever sense. If you are using the Darboux integral you should write so.2012-10-13
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    Very well. I will update the post2012-10-13
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    @PatrickDaSilva Ok any thoughts for the proof? Have I done a mistake or is the proof here http://math.berkeley.edu/~wodzicki/H104.F10/Integral.pdf incorrect2012-10-14

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Here is the full proof: Lemma: Let $I_1\subseteq \mathbb{R}^n$, $I_2\subseteq \mathbb{R}^m$ be boxes and $f:I_1\times I_2\to \mathbb{R}$ be bounded. Then, \begin{gather}\int_{I_1\times I_2*}f\le \int_{I_1*}\left(\int_{I_2*}f(x,y)\;dy\right)\;dx\le \left\{\begin{matrix} \displaystyle\int_{I_1}^*\left(\displaystyle\int_{I_2*}f(x,y)\;dy\right)\;dx\\ \displaystyle\int_{I_1*}\left(\displaystyle\int_{I_2}^*f(x,y)\;dy\right)\;dx\\ \end{matrix}\right\}\le \int_{I_1}^*\left(\int_{I_2}^*f(x,y)\;dy\right)\;dx\le \int_{I_1\times I_2}^*f \end{gather}

Proof: Observe that any partition $\Delta$ of $I_1\times I_2\subseteq \mathbb{R}^{n+m}$ can be expressed as $\mathcal{Q}\times \mathcal{R}$ where $\mathcal{Q},\mathcal{R}$ are partitions of $I_1$ and $I_2$ respectively. In addition, if for arbitrary $i=1,...,k_1, j=1,...,k_2$ we consider the product $\mathcal{Q}_{i}\times\mathcal{R}_{j}$, then we can create a partition $\mathcal{P}$ of $I_1\times I_2$ that is finer than $\Delta$ and $\mathcal{P}_{i,j}=\mathcal{Q}_{i}\times\mathcal{R}_{j}$

Therefore, \begin{gather}\inf_{(x,y)\in \mathcal{P}_{i,j}}f(x,y)\le f(x,y)\Rightarrow \inf_{(x,y)\in \mathcal{P}_{i,j}}f(x,y)vol(\mathcal{R}_j)= \int_{\mathcal{R}_j*}\inf_{(x,y)\in \mathcal{P}_{i,j}}f(x,y)\;dy\le \int_{\mathcal{R}_j*}f(x,y)\; dy\Rightarrow\notag\\ \inf_{(x,y)\in \mathcal{P}_{i,j}}f(x,y)vol(\mathcal{R}_j)vol(\mathcal{Q}_i)=\int_{\mathcal{Q}_i*}\inf_{(x,y)\in \mathcal{P}_{i,j}}vol(\mathcal{R}_j)\;dx\le \int_{\mathcal{Q}_i*}\left(\int_{\mathcal{R}_j*}f(x,y)\;dy\right)\;dx\Rightarrow\notag\\ \inf_{(x,y)\in \mathcal{P}_{i,j}}f(x,y)vol(\mathcal{P}_{i,j})\le\int_{\mathcal{Q}_i*}\left(\int_{\mathcal{R}_j*}f(x,y)\;dy\right)\;dx \end{gather} and so \begin{equation} L_{f,\Delta}\le L_{f,\mathcal{P}}=\sum\limits_{i=1}^{k_1}\sum\limits_{j=1}^{k_2}\inf_{(x,y)\in \mathcal{P}_{i,j}}f(x,y)vol(\mathcal{P}_{i,j})\le \sum\limits_{i=1}^{k_1}\sum\limits_{j=1}^{k_2}\int_{\mathcal{Q}_i*}\left(\int_{\mathcal{R}_i*}f(x,y)\;dy\right)\;dx =\int_{I_1*}\left(\int_{I_2*}f(x,y)\;dy\right)\;dx\end{equation} Taking the supremum with respect to all partitions $\Delta$ of $I_1\times I_2$ yields that: \begin{equation}\int_{I_1\times I_2*}f\le \int_{I_1*}\left(\int_{I_2*}f(x,y)\;dy\right)\;dx\end{equation} which is the first inequality of this Lemma. The middle two inequalites are obvious and the last inequality follows simillarly.