The Excision Property of the Lebesgue Measure

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:
(1)
\begin{align} \quad m(B) &= m(B \cap A) + m(B \cap A^c) \\ &= m(A) + m(B \setminus A) \end{align}
  • Since $m(A) < \infty$ we can subtract this quantity from both sides of the equation above to get:
(2)
\begin{align} \quad m(B \setminus A) = m(B) - m(A) \quad \blacksquare \end{align}
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