TITLE: Translation invariant measures on $\mathbb R$.
QUESTION [14 upvotes]: What are all the translation invariant measures on $\mathbb{R}$?
Except Lebesgue measure on $\mathbb R$ I didn't find any translation invariant measure. So I put this question?
I know that if $\mu$ is a measure then $c \times \mu$ is again a measure where $c>0$.

REPLY [4 votes]: Lebesgue measure and its multiples are not the only measures that are invariant under translations. Take, for example, the counting measure over the integers or over the rationals. Sure, they are infinite, but your question doesn't mention finiteness.