I start saying that I know this question has been already proposed a thousand times on this forum BUT I couldn't find a direct answer so I try asking on my own.  What's the intuitive meaning of this definition of measurable set?  $\cdot$ Let $X$ be a set and $\lambda^*$ an outer measure. Then $X$ is said to be measurable if for each $B\in X \lambda^*(B)=\lambda^*(B\cap X) + \lambda^*(B\cap X^C)$  Or equivalently if  $\lambda_*(X)=\lambda^*(X)$.  I feel OK with the second definition because it looks kind of similar to the intuition in Riemann Jordan "measure", but why are the two equivalent?  Thanks.
This week I saw the definition of a measurable set for an outer measure.     Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + \mu^*(A^c\cap E)$$ for every $E \subseteq X$.   This is not the first time I've seen this definition. Unlike most other things in mathematics, over time I have gained absolutely no intuition as to why this is the definition.   The only explanation I've ever seen is that a set is measurable if it 'breaks up' other sets in the way you'd want. I don't really see why this is the motivation though. One reason I am not comfortable with it is that you require a measurable set to break up sets which, according to this definition, are non-measurable; why would you require that? Of course, you can't say what a non-measurable set is without first defining what it means to be measurable so I suppose no matter what your condition is, it will have to apply to all subsets of $X$.     Is there an intuitive way to think about the definition of measurable sets? Is there a good reason why we should use this definition, aside from "it works"?
Why can we flag/vote a question on Meta as duplicate of another question which has no accepted/upvoted answers? When we flag/vote-to-close a question as a duplicate on Stack Overflow, and that potential duplicate doesn't have any upvoted or accepted answers, we are alerted that we can't use that question as a duplicate and can't continue. This same check does however not seem to apply to Meta questions. Why not? How to print a date in a regular format? Import/export 3ds format in Blender 2.8 I need to import and specially export in 3DS format. I know that I can import a file in 2.79 and the copy-paste to 2.8, but then when I copy-paste it back to 2.79 I'm having problems with textures missing. Is there any way to import/export 3ds in 2.8 without using any other version or software like we did in 2.79? The extension is missing in the add-ons repository either. Run cmd.exe in Administrator mode without explorer.exe I've recently had trouble with explorer.exe not working.  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There could also be data such as "abc", which should also throw exceptions.  Of course, there are countless variations of the examples I have posted, as well as the enormous complication of international phone numbers, which have quite complicated validation rules. This is why I would not consider rolling my own.  Has anyone seen such an API? Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from. How do berry mutations work? Sufficient Identification for Intra Schengen flights If one takes a direct flight from one Schengen member state to another, what's the minimum sufficient ID that needs to be provided to board the flight? 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