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If $f$ is integrable in $\mathbb{R}^d$ as for the Lebesgue measure and $\int_{R}f=0$ for each rectangle $R$, then $f=0$ almost everywhere.

Could you give me some hints how to show it??

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You can try the following step:

  • show that for each positive $\varepsilon$, there exists a $\delta$ such that if $B\subset\mathbb R^d$ is a Borel subset of Lebesgue measure smaller than $\delta$, then $\int_B|f|\mathrm d\lambda_d\lt\varepsilon$;
  • if $B$ is a Borel subset of finite measure, there exists a finite union of rectangles $B'$ such that $\lambda_d(B\Delta B')\lt\delta$. Assume first it is contained in $[-R,R]^d$ for some $R$, and use the fact that in a finite measure space, each measurable set can be approximated for the metric $d(A,B):=\mu(A\Delta B)$ by an element of a generating algebra. Here, you will need to use the algebra of finite unions of rectangles.
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  • $\begingroup$ Could you explain it further to me?? $\endgroup$ – Mary Star Nov 12 '14 at 10:39
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    $\begingroup$ Could you be more specific? Which step is problematic and why? $\endgroup$ – Davide Giraudo Nov 12 '14 at 12:22
  • $\begingroup$ I haven`t understood the second step... $\endgroup$ – Mary Star Nov 12 '14 at 12:27
  • $\begingroup$ @MaryStar I've edited. Is it clearer? $\endgroup$ – Davide Giraudo Nov 13 '14 at 10:58
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    $\begingroup$ I fail to understand how you intend the OP to approach your first step. $\endgroup$ – Did Nov 13 '14 at 11:02
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This follows from the Lebesgue differentiation theorem, if it is available. For every $x\in\mathbb R^n$, let $(R_k(x))_{k\ge0}$ be a sequence of rectangles with positive measure tending to $\{x\}$ (e.g. $R_k(x)=[x_1-1/k,x_1+1/k]\times\dots\times[x_n-1/k,x_n+1/k]$). Since the family of rectangles of $\mathbb R^n$ is of bounded eccentricity (in the sense of the Wikipedia page), for a.e. $x\in\mathbb R^n$, $$ f(x)=\lim_{k\to\infty}\frac1{\left|R_k(x)\right|}\int_{R_k(x)}f(t)\,\mu(\mathrm dt)=0. $$

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  • $\begingroup$ We haven`t got taught the Lebesgue differentiation theorem... How else could we show that?? In my notes there is the following: $$\int |f| d\mu =0 \Leftrightarrow f=0$$ could we maybe use that?? $\endgroup$ – Mary Star Nov 12 '14 at 10:37

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