Caratheodory says that $E\subset \mathbb R^d$ is measurable if $$m^*(A)=m^*(E\cap A)+m^*(E^c\cap A)$$ for all $A\subset \mathbb R^d$. To me a more natural way would have been to define it as : $E$ is measurable if $$m^*(E)=m^*(E\cap A)+m^*(E\cap A^c)$$ for all $A\subset \mathbb R^d$. But, maybe the equality hold for all $A\subset \mathbb R^d$, even if $E$ is not measurable, that's why it wouldn't be a good definition. What do you think ?
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Your first equation should say $$m^(A)=m^(E\cap A)+m^*(E^c \cap A).$$ – kccu May 20 '16 at 17:36
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@kccu: I corrected it, thanks, – MSE May 20 '16 at 17:38
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1Let $E = [0,1]$. Clearly $E$ is measurable but for certain nonmeasurable sets $A$ it is possible that $m^(E \cap A) = m^(E \cap A^c) = 1$. – Umberto P. May 20 '16 at 17:42