I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of the Riesz representation theorem: if we had a linear, translation-invariant functional that satisfied a positivity property, by the Riesz representation theorem we'd get our measure. To that end he constructs, for any $f_1, \ldots, f_n \in \mathcal{C}_c(G)^+$ and $\varepsilon > 0$, a functional $I = I_{f_1, \ldots, f_n, \varepsilon}$ that is
Homogeneous,
"Almost linear" on the functions we've chosen in the sense that $|I(f_i + f_j) - I(f_i) - I(f_j)| < \varepsilon$,
Translation-invariant,
Nontrivial in the sense that it is nonzero on a predetermined nonzero $f_0$, and
Uniformly bounded (in the $f_i$ and $\varepsilon$) by a sublinear functional $K$.
He then sketches a couple different arguments to show the existence of the functional that we want from here.
One of the arguments he talks about uses the Hahn-Banach theorem. The gist, he says, is
If one lets $\mathcal C$ be the space of all tuples $(f_1,\ldots,f_n,\epsilon)$, one can use the Hahn-Banach theorem to construct a bounded real linear functional $\lambda: \ell^\infty({\mathcal C}) \rightarrow {\mathbb R}$ that maps the constant sequence $1$ to $1$. If one then applies this functional to the $I_{f_1,\ldots,f_n,\epsilon}$ one can obtain a functional $I$ with the required properties.
Could someone explain this argument or point me to a source where it's worked out in more detail? I can't even get started on how this would go.