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I started studying "Measure Theory and Integration" and went through the first section which talks about Lebesgue Outer Measure of a set. All was well until I started with the second section which starts with the definition of Measurable sets :

** The set $E$ is Lebesgue Measurable if for each set $A$ we have**

$m^*(A)=m^*(A\bigcap E)+m^*(A\bigcap E^c)$

I know the $\leq$ inequality comes from subadditivity.

So it all boils down to showing the $\geq$ inequality.

Though I read the definition and am able to solve questions on the topic. I don't quite understand it intuitively. Any help would be appreciated.

Also give me an example where $\geq$ inequality is not satisfied.

Thank you.

drhab
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Swapnil Tripathi
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  • I would say that measurable sets are sets that are split up by any other set on a 'nice' way. If there is more to find on intuition then I will learn from that too. Asking for an example where $\geq$ is not satisfied is the same as asking for a set that is not Lebesgue-measurable. – drhab Sep 06 '14 at 09:57
  • I don't know any examples. Can you please tell me one? – Swapnil Tripathi Sep 06 '14 at 10:27
  • I haven't got them 'in my pocket'. Maybe this can help. – drhab Sep 06 '14 at 10:32
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    There are different (equivalent) definitions of a measurable set. The one you have showed is named after Caratheodory. Maybe this will have what you search for, regarding intuition.

    Also, though there are non-lebesgue measurable sets, they are extremely rare, with complex construction, so I don't think they will provide much intuition. For the most part, most sets that you "encounter" are measurable.

     – Aahz Sep 06 '14 at 10:41

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