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Problem:

$f$ is Lebesgue integrable on $E\subset\mathbb{R}$, what is $I=\{\int_e fdm: \text{Measurable set}\ e\subset E\}$ ?

I guess the answer is $[\int_{E_1}fdm,\int_{E_2}fdm]$ where $E_1=\{x\in E: f(x)<0\}$ $E_2=\{x\in E: f(x)>0\}$. I also think that the following property of Lebesgue integral might be of help, but I don't know how to use this property to solve this problem...

Absolute continuity of the Lebesgue integral: $f$ is an Lebesgue integrable function on $E$, $\forall\epsilon>0\ \exists\delta>0$ such that for any measurable set $E_0\subset E$, if $m(E_0)<\delta$, then $\vert\int_{E_0}fdm\vert<\epsilon$.

Guanfei
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    This is correct. First, it is clear that $I\subset [\int_{E_1} f, \int_{E_2} f]$. Now try integrating over the sets $[\inf E, x]\cap E_1$ and $[\inf E, x]\cap E_2$ (note if $E$ is for instance a subset of $\mathbb R^n$ then the argument needs to be modified.) – Maximilian Janisch Sep 05 '20 at 15:19
  • On what set is your $f$ defined? Is $E \subseteq \mathbb R$? – mathcounterexamples.net Sep 05 '20 at 15:40
  • Right. It is defined on $\mathbb{R}$. – Guanfei Sep 05 '20 at 15:41
  • @Guanfei Great, then my argument works – Maximilian Janisch Sep 05 '20 at 15:44
  • @Janisch: Thanks! After integrating over $[infE,x]\cap E_1$, I can see that $\int_{E_1}\leq\int_{[infE,x]\cap E_1}fdm<0$. I think the next step is that when $x$ moves towards infinty, any values between $\int_{E_1}fdm$ and $0$ can be achieved by $\int_{[infE,x]\cap E_1}fdm$. But how can we prove this, i.e. Lebesgue integral is continous with regard to its domain of integration? I know this may be a basic question, but I do not have a solid foundation in analysis... – Guanfei Sep 05 '20 at 15:50
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    Compare https://math.stackexchange.com/a/2412182/42969 – Martin R Sep 05 '20 at 16:03
  • @Martin: Thank you! I thought the Absolute continuity of the Lebesgue integral can be used, but in fact Lebesgue's dominated convergence theorem is used instead. – Guanfei Sep 05 '20 at 16:14
  • @Martin: I just came up with another thought: since Lebesgue's dominated convergence theorem only applies when there is a dominating integrable function, does it mean that "Lebesgue integral is continuous with regard to its domain of integration" is not always true? – Guanfei Sep 05 '20 at 16:27
  • @Guanfei My idea would be to prove that $$x\mapsto \int_{[\inf E, x]\cap E_1} f,\mathrm dm$$ is a continuous function using absolute continuity of the Lebesgue integral. Then conclude (let $x$ go to $\inf E$ and let $x$ go to $\infty$.) Same thing for $E_2$. – Maximilian Janisch Sep 05 '20 at 18:03

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