I´ve been reading about amenability of groups, and I don´t know if the definition involving the existence of $\mu$ a finitely additive, left-invariant mean on $\mathbb{B}(G)$(the Borel set on G) with $\mu(G)=1$ is equivalent to the definition by the existence of left invariant mean on $L^{\infty}(G)$ when G is locally compact.
Is there any locally compact group, amenable(in the mean sense) that is not amenable in the invariant measure sense.
Thanks
In fact, the two definitions you mentioned are equivalents. This because the dual of $L^\infty (G)$ is the vector space $\mathcal{M}(G)$ of all finitely additive finite signed measures defined on the Borel algebra of $G$. See here. You can check that the map between the two vector spaces is $G$-equivariant and so once you know that there is an invariant mean on $G$ you can construct a finitely additive, left-invariant Borel probability measure on $G$ and viceversa.