I am trying to show two sets are equal.
Basically I have a set that is outer measurable, i.e. it satisfies the following:
$m^{*}(A) \geq m^{*}(A \cap E) + m^{*}(A \cap E^c)$ where $E$ is a measurable set and $A$ is an set in $R$. Basically I only consider the real space.
I am asking this question because I want to show that even if $E$ is being translated say by $E + y$ where $y$ is a number, I still could have $E+y$ being Outer-measurable.
My question is this:
how do I show that $A \cap (E+y)^c = ((A-y) \cap E^c) +y$
I want to have that because then, I can say:
$m^{*}(A \cap (E+y)^c) = m^*( ((A-y) \cap E^c) +y ) )$
and hence also the left equal $m^*( ((A-y) \cap E^c) ) )$
because outer-measure is translation invariant.
It is the complement of the $E+y$ and $E$ that is bugging me because for the other part, I was able to show the following two sets are equal:
$A \cap (E+y)$ = $((A-y) \cap E)+ y$
Basically it is the Page 5 in this pdf: https://www.csie.ntu.edu.tw/~b89089/book/Royden/Royden_3.pdf
Just follow the steps they had but with $\not\in$:
$x\in A\cap (E+y)^c\iff \left(x\in A \right)\wedge \left(x\not\in E+y\right),\\ \iff \left(x\in A \right)\wedge \left(x-y\not\in E\right),\\ \iff \left(x-y\in A-y \right)\wedge \left(x-y\not\in E\right)\\ \iff x-y\in (A-y)\cap E^c\\ \iff x\in \left((A-y)\cap E^c\right)+y$