How do I prove the following?
If X is a Borel set, and $\displaystyle a \in \mathbb{R}$, prove that $\displaystyle X+a=\{x+a, x \in X\}$ is also a Borel set.
Thank you!
When you show that the generators (pi systems) are the same. So in this case the Borel sets are generated by the open sets. As $\displaystyle \cup(A_i+a)=\cup(A_i)+a$, the sets of the form X+a are generated by U+a where U is open. If you know that the generators are the same, then you know that the sigma algebra they generate are the same as well.