If $A$ and $B$ are measurable, then there is an equality: $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B)$. But is $\lambda^*(A \cap B) + \lambda^*(A \cup B) \leq \lambda^*(A) + \lambda^*(B)$ true for any sets $A$ and $B$.
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What is $\lambda^$? If $A,B$ are not measurable then is $\lambda^ (A)$ even defined? – SL_MathGuy Jan 11 '20 at 19:53
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@SL_MathGuy I think $\lambda^*$ is the outer measure, and a non-measurable set can have an outer measure. – Botond Jan 11 '20 at 19:55
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@SL_MathGuy $A$ and $B$ are any sort of sets, even non-measurable. Outer measure defined even for non-measurable sets. – A.Kat Jan 11 '20 at 20:12
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3Does this answer your question? For an outer measure $m^$, does $m^(E\cup A)+m^(E\cap A) = m^(E)+m^*(A)$ always hold? – SL_MathGuy Jan 11 '20 at 21:42
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@SL_MathGuy in this question one of sets is measurable. – A.Kat Jan 11 '20 at 22:06