Lots of definitions follow, the question is further down.

Let $\Gamma(\mu_1,\mu_2)$ be the set: $$\Gamma(\mu_1,\mu_2) = \{ \gamma \text{ probability measure on } \mathbb{R}^d \times \mathbb{R}^d \text{ with }\mu_1\text{ and }\mu_2\text{ as marginals }\}$$

The Primal Kantorovich Problem is to find $\gamma \in \Gamma(\mu_1, \mu_2)$ which minimizes the functional $C_K[\gamma]$ which is explicitly given as: $$\inf_{\gamma \in\Gamma(\mu_1,\mu_2)}C_K[\gamma] = \int_{\mathbb{R}^d \times \mathbb{R}^d}c(x,y) \gamma(dx,dy)$$ where $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ has some nice properties (i.e. lsc + other good stuff).

The way this problem is formed, it is a linear function over a convex set. So duality theory from convex analysis can be used to analyze the existence and uniqueness of this problem's minimizer. I am reading a set of notes that uses the following procedure to obtain a formulation of the dual statement:

Define an indicator function $$ \mathbf{1}(\gamma) = \begin{cases} 0 & \gamma \in \Gamma(\mu_1,\mu_2) \\ + \infty & \gamma \notin \Gamma(\mu_1,\mu_2) \end{cases} $$ Then express this indicator function somehow over $C_0(\mathbb{R}^d) \times C_0(\mathbb{R}^d)$. To do this write: $$\mathbf{1}(\gamma) = \sup_{(u,v) \in C_0(\mathbb{R}^d) \times C_0(\mathbb{R}^d)}L_\gamma(u,v)$$ where $$L_\gamma(u,v) = \int_{\mathbb{R^d}}u(x) \,d\mu_1(x) - \int_{\mathbb{R^d} \times\mathbb{R}^d}u(x) \,d\gamma(dx,dy) + \int_{\mathbb{R^d}}v(y) \,d\mu_2(y)- \int_{\mathbb{R^d} \times \mathbb{R}^d}v(y) \,d\gamma(dx,dy)$$

My question:

Presumably, there are a variety of ways to express this indicator function. Why should we choose $C_0(\mathbb{R}^d) \times C_0(\mathbb{R}^d)$ as the set over which to express the indicator function? In particular, if I had chosen a different set, I would obtain a different dual problem right? Does this mean there are many dual problems for a given primal problem?


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