I'm reading a paper ("A simple proof in Monge-Kantorovich Theory", by D. A. Edwards), and at a very crucial point the author's applying Hahn-Banach in a way I don't understand. Appreciate every help!
This is Hahn-Banach as I know it:
Let V be a vector space, $U$ a linear subspace, $l: U \to \mathbb{R}$ a linear functional and $p:V\to\mathbb{R}$ a sublinear functional with $l(u) \leq p(u) ~ \forall u\in U$.
Then there exists a linear function $L: V\to \mathbb{R}$ such that:
$$
L(u) = l(u) ~ \forall u \in U, \qquad L(v) \leq p(v) \forall v \in V.
$$
This is the author's application:
Let $X, Y$ be completely regular spaces and $Z:= X \times Y$.
For an upper semicontinuous function $f$ on $Z$ and two Borel measures $\mu$ (on X) and $\nu$ (on Y) we define
$$
\phi(f):= \{(u,v)|~~ u:X\to (-\infty, \infty], v:Y\to (-\infty, \infty], u\in \mathcal{L}^1(\mu), \nu \in \mathcal{L}^1(\nu), f(x, y) \leq u(x) + v(y) ~ \forall x, y \in Z\}.
$$
and the sublinear functional $p$ at $f$ by
$$
p(f) := \inf\{\mu(u)+\nu(v): (u, v) \in \phi(f)\}
$$
(this is indeed sublinear).
Now let $h \in C_b(Z)$ (real-valued, bounded, continuous functions). The author claims: "By the Hahn-Banach theorem there exists a linear functional $I: \mathcal{C}_b(Z) \to \mathbb{R}$ such that $$ I(f) \leq p(f) ~ \forall f\in \mathcal{C}_b(Z), \qquad I(h)=p(h). $$
What I do not understand: what would be the linear functional (called $l$ in my quote of Hahn-Banach) and the linear subspace $U$? And how come we have equality at $h$?