I started studying "Measure Theory and Integration" and went through the first section which talks about Lebesgue Outer Measure of a set. All was well until I started with the second section which starts with the definition of Measurable sets :
** The set $E$ is Lebesgue Measurable if for each set $A$ we have**
$m^*(A)=m^*(A\bigcap E)+m^*(A\bigcap E^c)$
I know the $\leq$ inequality comes from subadditivity.
So it all boils down to showing the $\geq$ inequality.
Though I read the definition and am able to solve questions on the topic. I don't quite understand it intuitively. Any help would be appreciated.
Also give me an example where $\geq$ inequality is not satisfied.
Thank you.