Let $\gamma$ be a measure on $\mathscr{B}(\mathbb{R})$, invariant under translations, i.e. $\gamma(B+b) = \gamma(B)$ for all $B\in\mathscr{B}(\mathbb{R})$ and $b\in\mathbb{R}$,$\hspace{0.2cm}$ and $\hspace{0.2cm}$ $\gamma(\mathbb{R})< \infty$. Prove that $\gamma \equiv 0$.
I've tried to solve this but I can't. I know that setting $k:=\gamma((0,1))$, because measure is invariant under transalations I have $\gamma((r,r+n))=\gamma((0,n)+r)=\gamma((0,n))=nk$ for all $r\in\mathbb{R}$ and $n\in\mathbb{N}$ and maybe from this I could get an useful equality or inequality.
Another thing to consider is the hypothesis $\gamma(\mathbb{R})<\infty$, I don't know how to use it. Any ideas is welcome, thanks.
Well, $$ \sum_{n=-\infty}^{\infty}\gamma([n,n+1))=\sum_{n=-\infty}^{\infty}\gamma([0,1)), $$ and the RHS should be finite.