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There is a standard definition for measure theory which defines measurable sets for an outer measure $\mu^*$ as those which have the property that:

$\mu(E) = \mu(E\cap A) + \mu(E \cap A^c)$

My question is, why should we require the measurable sets behave well with all sets instead of those in a strictly smaller subset?

For example, why not restrict $E$ to be an element of the borel sigma algebra instead of a general subset of $R$?

maxical
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  • The definition with the Borel Sigma algebra relies on the topological properties of the space, i.e., one needs to make sense of Borel sets on a given space. Making measures play nice with all sets generalizes the theory. – rubikscube09 Apr 30 '20 at 18:00
  • I understand the idea of generalization but, the question still stands - to what extent can more assumptions be added and still get the core result of measure theory on $R$ which is the MCT and DCT? – maxical Apr 30 '20 at 18:03
  • Take a look at https://math.stackexchange.com/questions/485815/intuition-behind-the-definition-of-a-measurable-set and the questions which link to it. – Nate Eldredge Apr 30 '20 at 19:18

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