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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

David Ariza
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    Could you edit your post? There are sentences which seem you're finishing without remembering how you started them. And I have little idea what you're asking. – YCor May 08 '19 at 21:54

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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.

Evian
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