I am supposed to show that if C is a lebesgue measurable set, then C+x is lebesgue measurable, and $\lambda(C)=\lambda(C+x)$.
The last one is ok, because I have that their outher measures must agree, since they are created by taking the inf over open coverings of intervals, and since when you translate an interval, it keeps it length, we take the inf over the same set.
My problem is showing that it is lebesgue measurable. The only way I see of doing that is to show that it satisfies the caratheodory criterion. I need to show that for every set W:
$\lambda^*(W)=\lambda^*(W\cap(C+x))+\lambda^*(W \cap (C+x)^c)$
Since the outer measure satisfies subadditivity it is eough to show that:
$\lambda^*(W)\ge\lambda^*(W\cap(C+x))+\lambda^*(W \cap (C+x)^c)$.
How am I supposed to show this though? I am thinking about translating W and then using the measurability of C, so the first one becomes $\lambda^*(W\cap(C+x))=\lambda^*((W-x)\cap(C))$, if I could do the same with the second one I am done I think. However in order to do that I must have that $(C+x)^c-x=C^c$, but is this true?, and how do you show that? If not is there another way to do this?
The following statements are equivalent:
$y\in\left(C+x\right)^{c}-x$
$y+x\in\left(C+x\right)^{c}$
$y+x\notin C+x$
$y\notin C$
$y\in C^{c}$