The Excision Property of the Lebesgue Measure
Recall from The Lebesgue Measure page that the Lebesgue measure $m$ is a set function defined on the set of all Lebesgue measurable sets $\mathcal M$ that is identically the Lebesgue outer measure set function restricted to $\mathcal M$.
We now prove an important property of $m$ - the excision property.
|Theorem 1 (The Excision Property of the Lebesgue Measure): Let $A$ and $B$ be Lebesgue measurable sets with $A \subseteq B$ and $m(A) < \infty$. Then $m(B \setminus A) = m(B) - m(A)$.|
- Proof: Since $A$ and $B$ are Lebesgue measurable we have that:
- Since $m(A) < \infty$ we can subtract this quantity from both sides of the equation above to get: