I am reading a paper, and I got stuck at an implication, which the authors believed to be natural, and hence did not provide a reasoning. Unfortunately, I can not see immediately why that implication is true. Any kind of help/comments are greatly appreciated. The problem with the implication is the following.
Let $(S, \cdot)$ be a locally compact semigroup, and $C_b(S)$ denote the space of all bounded continuous functions on $S$. For any $s\in S, f\in C_b(S) $, the left translate $L_sf\in C_b(S)$ is defined as $L_sf(t):=f(s.t)$ for each $t\in S$.
A left-invariant mean on $S$ is a functional $m\in C_b(S)^* $ such that $||m||=m(\chi_S)=1$ and $m(L_sf)=m(f)$ for each $s\in S, f\in C_b(S) $, where $\chi_S$ is the characteristic function on $S$, i.e, the constant function $\mathbb{1}$.
It is given/proved that there exists a linear functional $m\in C_b(S)^* $ such that $m(\chi_S)=1$ and $m(L_sf)=m(f)$ for each $s\in S, f\in C_b(S) $.
Question: How can we construct a left-invariant mean on $S$, using $m$ ?
A Hint was given to me in the lines of decomposing $m$ into positive and negative parts. I tried using certain manipulations as well, but could not succeed in that. (I understand that it is enough to find a suitable positive linear functional that takes the value $1$ at $\chi_S$.)
Thank you in advance, for the help!