I am reading Brooks and Series, Bounded Cohomology for Surface Groups.
At the end of the first paragraph on the last page, it says "the homomorphisms $\Gamma\to\mathbb{R}$ are linear combinations of the functions $f_g$, $g\in\Gamma_0$", where $\Gamma_0=\{a_i,b_i,a_i^{-1},b_i^{-1}\}_{i=1}^g$ is the set of generators of the surface group and $f_g=2(h_g-h_{g^{-1}})$. Here $h_w(x)$ means counting the word $w$ inside the word $x$ (to my understanding).
I have problem understanding this... Is there a proof of this fact?