What is the difference between outer measure and inner measure ?
From the definition, the outer measure must be bigger, but under which conditions are they equal, for example for measurable $A$ and any $B\supset A$, does it always follow that;
$$m^*(B\cap A)=m(A)$$
or Is there an equation, for these 2 measures ?
I have 2 Examples
$1.$ If $m(A)=0$, show that for any $B$
$m^*(A\cup B)=m^*(B\setminus A)=m^*(B)$
Author's solution:
$m^*(B)\le m^*(A\cup B)=m^*(B\setminus A)+m^*(A)\le m^*(B)+m^*(A)\overset?=m^*(B)$
(So $m(A)=0\Rightarrow m^*(A)=0$, but in general the converse of this implication is true.)
$2.$ Show if $A$ is measurable and a set of finite measure and $B\supset A$
$m^*(B\setminus A)=m^*(B)-m(A)$
Author's solution:
Carathéodory-criterion implies;
$m^*(B)=m^*(B\cap A)+m^*(B\setminus A)\overset?=m(A)+m^*(B\setminus A)$
(He assumed $m^*(B\cap A)=m(A)$, but why ?)
Best Answer
The outer measure is monotonic, and $A \subset B \implies A= B \cap A$, and therefore: $$m^{*}(B \cap A) = m^{*}(A)$$ Since $A$ is measurable, $m(A)=m^{*}(A)$, and finally: $$m^{*}(B \cap A) = m(A)$$