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Here, E is a Lesbegue-measurable set on the real line. This is the exercise 30, 31 of p. 40 of Folland real analysis. I solved these problems when E is of finite measure, but the problem requires that E may be of infinite measure. I'm quite desperate about how to solve these for general cases. Could anyone show me how to prove them?

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    $\begingroup$ Note that $\mathbb R=\bigcup_{n\in\mathbb Z}(n,n+1]$. Since $m(E)>0$, it must be the case that $m(E\cap(n,n+1])>0$ for some $n\in\mathbb Z$. But $m(E\cap(n,n+1])\leq m((n,n+1])=1$, so the problem may be effectively reduced to the case in which $m(E)<\infty$, which you claim to have already proved. $\endgroup$ – triple_sec Mar 26 '15 at 16:24
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    $\begingroup$ This is essentially a well known result known as Steinhaus lemma. $\endgroup$ – Batman Mar 26 '15 at 16:41
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this will be the desired solution for problem 30.

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    $\begingroup$ write the answer in a compact form using MathJax $\endgroup$ – vidyarthi Feb 12 '17 at 10:06
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    $\begingroup$ I am sry brother but I don't know how to use it and how to write in it, that's why I wrote on a piece of paper and attached the picture of the proof $\endgroup$ – bunny Feb 12 '17 at 10:23
  • $\begingroup$ see here for more details and use it next time. $\endgroup$ – vidyarthi Feb 12 '17 at 11:39

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