Excision property of measurable sets (Proof)

If $A$ is a measurable set of finite outer measure that is contained in $B$, then $\displaystyle m^*(B\setminus A)=m^*(B)-m^*(A).$

Proof:

By the measurability of $A$,
\begin{aligned} m^*(B)&=m^*(B\cap A)+m^*(B\cap A^c)\\ &=m^*(A)+m^*(B\setminus A). \end{aligned}
Since $m^*(A)<\infty$, we have the result.

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