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A set $A$ is measurable if for any subset $E$ of $\Bbb R$ the following holds:

$$ m_{*}(E \cap A) + m_{*}(E \cap A^c) = m_{*}(E).$$ What is the motivation behind such definition ?

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  • $\begingroup$ From Hewitt and Stromberg Real and Abstract Analysis: "How Carathéodory came to think of this definition seems mysterious, since it is not in the least intuitive. Carathéodory's definition has many useful implications. It gives us a $\sigma$-algebra ... on which $\mu$ is a countably additive measure." $\endgroup$ – David Mitra Feb 7 '14 at 14:23
  • $\begingroup$ It made sense to me once I read through the definitions keeping in mind that our goal is to approximate the 'volume' of a set using open sets around the set we are interested in. $\endgroup$ – mathematics2x2life Feb 7 '14 at 14:27
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(Assuming $m_*$ is outer measure.) There is some information on this in a note included in Carathéodory's collected works. It was previously known for Lebesgue measure in the real line, that a set $A$ is measurable if and only if $$ m_{*}(E \cap A) + m_{*}(E \cap A^c) = m_{*}(E) $$ for all intervals $E$. When Carathéodory wanted to do an abstract version of the theory (not necessarily Lebesgue measure), it is either nonsense (or of no use) to use intervals here. Then he discovered that this same criterion, required for all sets, not just intervals, also holds for Lebesgue measure; so that could be used in the abstract setting.

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