How to prove that some set is a Borel set If $B$ is a borel set, is $B+c$ a borel set for some constant $c$ ? I know that it is not possible to characterize a Borel set. 

Yes. You know that the set $B+c$ is the inverse image of the measurable set, $B$, under the continuous function:  $f(x)=x-c$, and since continuous functions are measurable, and the inverse image of a measurable set is measurable, $B+c$ is also measurable.

Note: there is nothing particularly special about Lebesgue measure here other than it being a Haar measure. I only use that the measure is a Haar measure on a locally compact group:  in any such case translation is a continuous operation, so you can get the same result.
