Excision Property in Measure Theory

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.

Unknown's avatar

Author: mathtuition88

Math and Education Blog

Leave a comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.