Let $\mathcal{B}$ Borel algebra, $\lambda$ Lebesgue measure in $\mathcal{B}$. I need to show that
(i) $\lambda(G)>0$ for all $G$ open.
(ii) $\lambda(K)<\infty$ for all $K$ compact.
(iii) $\lambda(x+E)=\lambda(E)$ for all $E\in\mathcal{B},$ where $x+E=\{x+y:y\in E\}.$
I proved the (i) and (ii) statements.
(i) $G$ open $\implies$ exists an open interval $I\subset G$. Since $\lambda(I)>0,$ $\lambda(G)>0.$
(ii) $K$ compact $\implies K$ bounded $\implies K$ is contained in an open interval $K\subset I$. Since $\lambda(I)<\infty\implies\lambda(K)<\infty.$
Now, the statement (iii) is really intuitive. The set $x+E$ is the set $E$ translated by $x$, so the Lebesgue measure needs to be fixed in this translation, but I don't know how to prove this for a general set $E\in\mathcal{B}.$
If $E$ is an open/closed interval is easy, because $$\lambda\left((a,b)\right)=\lambda(x+a,x+b)=b-a.$$
But for a general set $E\in\mathcal{B},$ I don't know how to do it.
What can I do?
Best Answer
When $E$ is a finite disjoint union of intervals of the type $(a,b]$ you can verify the equality as in the case of one open interval. You then have to complete the proof using the uniqueness part of Caratheodory Extension Theorem. Ref: https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension_theorem