Exercise 6, Appendix 0

Let $S$ be a set on which a group $\Gamma$ operates on the left (A, I, § 5, No. 1). Let $E$ be the real vector space $\mathscr{B}(S)$ of all bounded numerical functions on $S$ (I, p. 4, Example). Suppose that the group $\Gamma$ (endowed with the discrete topology) has a left invariant mean, and let $\Gamma$ operate on $E$ in such a way that $sf(x) = f(s^{-1}x)$ for all $s \in \Gamma$, $f \in E$ and $x \in S$.

Let $g \in E$ be a positive function; let $E_1$ be the vector subspace of $E$ generated by the functions $sg$, where $s$ ranges over $\Gamma$; let $E_2$ be the vector subspace of $E$ generated by the positive functions which are bounded by functions of $E_1$.

Show that if there exists a non null positive linear form $\phi$ on $E_1$, which is invariant under $\Gamma$, then there exists a non null positive linear form on $E_2$, which is invariant under $\Gamma$. (Using prop. 1 of II, p. 21, first construct a positive linear form on $E_2$, which extends $\phi$, and let $\Gamma$ operate on the set of these extensions). In particular consider the case where $g = 1$.