TITLE: Finitely additive measure on $\mathbb R$
QUESTION [8 upvotes]: Suppose $\mathcal B$ is the Borel $\sigma$-algebra on $\mathbb R$. Let $\mu : \mathcal B \rightarrow [0, \infty ]$ be a finitely additive(but not necessarily countably additive), translation-invariant, ''measure'', with the property that $\mu(K) < \infty$ for every compact $K \subset \mathbb R $. 
Let $P(\mathbb R)$ denote the power set of $\mathbb R$.
Does there exist a finitely additive, translation-invariant "measure", $\tilde \mu : P(\mathbb R) \rightarrow [0, \infty ]$, extending $\mu$ ?
In particular, let $\mu$ be the Lebesgue measure on the Borel $\sigma$-algebra. Can something be said about this special case at least?

REPLY [2 votes]: I don't know about the general case. But here Tao states that the Lebesgue case is true, using the Hahn-Banach theorem.