I'm reading the proof of Kantorovich duality from Villani's book *Topics in Optimal Transportation*.


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Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures on $X,Y$ respectively. Let $c: X \times Y \rightarrow \mathbb{R}_{\ge 0} \cup\{+\infty\}$ be lower semi-continuous. Fix $\mu \in P(X)$ and $\nu \in P(Y)$.

- $\Pi(\mu, \nu)$ is the set of $\pi \in P(X \times Y)$ such that for all measurable subsets $A \subset X$ and $B \subset Y$,
$$
\pi[A \times Y]=\mu[A], \quad \pi[X \times B]=\nu[B].
$$

- $\Phi_{c}$ is the set of all $(\varphi, \psi) \in L^{1}(d \mu) \times L^{1}(d \nu)$ satisfying
$$
\varphi(x)+\psi(y) \leq c(x, y)
$$
for $\mu$-almost all $x \in X$ and $\nu$-almost all $y \in Y$.

- For $\pi \in P(X \times Y)$ and $(\varphi, \psi) \in L^{1}(d \mu) \times L^{1}(d \nu)$, let $$I[\pi]:=\int_{X \times Y} c d \pi, \quad J(\varphi, \psi):=\int_{X} \varphi d \mu+\int_{Y} \psi d \nu .$$

Then
$$
\inf _{\Pi(\mu, \nu)} I[\pi]=\sup _{\Phi_{c}} J(\varphi, \psi) .
$$


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The author said that
>We separate the proof into three steps, by increasing order of generality. The minimax principle will only be applied in the first step, which is the case when $X$ and $Y$ are compact and $c$ is **continuous**. All the rest of the proof will consist in showing that this particular case implies the general statement, by approximation arguments.


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In the proof below, I could not see how the author uses the **continuity** of $c$. Could you elaborate on my confusion?

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**Proof:** It's clear that
$$
\sup _{\Phi_{c} \cap C_{b}} J(\varphi, \psi) \leq \sup _{\Phi_{c} \cap L^{1}} J(\varphi, \psi) \leq \inf _{\Pi(\mu, \nu)} I[\pi].
$$

So it remains to prove 
$$
\inf _{\Pi(\mu, \nu)} I[\pi] \le \sup _{\Phi_{c} \cap C_{b}} J(\varphi, \psi) .
$$

To simplify notations, let $\varphi \oplus \psi: (x,y) \mapsto \varphi(x)+\psi(y)$. We have
$$
\inf _{\pi \in \Pi(\mu, \nu)} I[\pi]=\inf _{\pi \in M_{+}(X \times Y)}\left(I[\pi]+\left\{\begin{array}{l}
0 \text { if } \pi \in \Pi(\mu, \nu) \\
+\infty \text { else }
\end{array}\right)\right.
$$
with $M_+(X \times Y)$ the space of non-negative Borel measures on $X\times Y$. Also,
$$
\left\{\begin{array}{l}
0 \text { if } \pi \in \Pi(\mu, \nu) \\
+\infty \text { else }
\end{array}\right\}=\sup _{(\varphi, \psi)} \left[\int \varphi d \mu+\int \psi d \nu-\int \varphi \oplus \psi d \pi\right],
$$
where the supremum on the RHS runs over all $(\varphi, \psi) \in C_b(X) \times C_b(Y)$. It follows that
$$
\begin{aligned}
 \inf _{\pi \in \Pi(\mu, \nu)} I[\pi]
 =\inf _{\pi \in M_{+}(X \times Y)} \sup _{(\varphi, \psi)}  \bigg \{ \int_{X \times Y} c d \pi +\int_{X} \varphi d \mu+\int_{Y} \psi d \nu - \int_{X \times Y} \varphi \oplus \psi d \pi \bigg\}.
\end{aligned}
$$

1. Let us first assume that $X, Y$ are compact and that $c$ is **continuous** on $X \times Y$.

- Let $E:=C_{b}(X \times Y)$ be the set of all bounded continuous functions on $X \times Y$, equipped with its usual supremum norm $\|\cdot\|_{\infty}$.

- By Riesz' theorem, the topological dual $E^*$ of $E$ can be identified with the space of regular Radon measures, $M(X \times Y)$, normed by total variation.
- Moreover, a nonnegative linear form on $E$ corresponds with a regular nonnegative Borel measure.

Then we introduce
$$
\begin{gathered}
\Theta: u \in E \longmapsto\left\{\begin{array}{l}
0 &\text {if } u \geq-c, \\
+\infty &\text {else}.
\end{array}\right.\\
\text{}\\
\Xi: u \in E \longmapsto\left\{\begin{array}{l}
\int_{X} \varphi d \mu+\int_{Y} \psi d \nu &\begin{align*}
&\text {if } u = \varphi \oplus \psi \text{ for}\\
&  \text{some } (\varphi, \psi) \in L^{1}(d \mu) \times L^{1}(d \nu)
\end{align*},\\
+\infty &\text {else. }
\end{array}\right.
\end{gathered}
$$

It's easy to verify that $\Theta, \Xi$ satisfy the condition of [Fenchel-Rockafellar](https://en.wikipedia.org/wiki/Fenchel%27s_duality_theorem#Mathematical_theorem) duality, so
$$
\inf _{u\in E}[\Theta(u)+\Xi(u)] = \max _{\pi \in E^{*}}\left[-\Theta^{*}\left(-\pi\right)-\Xi^{*}\left(\pi\right)\right].
$$

It's clear that
$$
\begin{align*}
\inf _{u\in E}[\Theta(u)+\Xi(u)] &= \inf \left\{\int_{X} \varphi d \mu+\int_{Y} \psi d \nu \,\middle\vert\, (\varphi, \psi) \in L^{1}(d \mu) \times L^{1}(d \nu)\text{ s.t. } \varphi \oplus \psi \geq -c \right\} \\
&=-\sup \left\{J(\varphi, \psi) \mid (\varphi, \psi) \in \Phi_{c}\right\}.
\end{align*}
$$

Next, we compute the Legendre-Fenchel transforms of $\Theta, \Xi$. First, for any $\pi \in E^*$,
$$
\begin{aligned}
\Theta^{*}(-\pi) =\sup _{u \in E}\left\{-\int u d \pi \,\middle\vert\, u \geq-c\right\}
= \sup _{u \in E}\left\{\int u d \pi \,\middle\vert\, u \leq c\right\} .
\end{aligned}
$$
- If $\pi$ is not nonnegative, then there exists a positive function $v \in E$ such that $\int v d \pi<0$. Then, the choice $u=\lambda v$, with $\lambda \rightarrow-\infty$, shows that the supremum is $+\infty$.
- On the other hand, if $\pi$ is nonnegative, then the supremum is clearly $\int c d \pi$.

Thus
$$
\Theta^{*}(-\pi) = \begin{cases}
\int c d \pi &\text {if } \pi \in M_{+}(X \times Y) \\
+\infty &\text {else}.
\end{cases}
$$

We also have
$$
\begin{align*}
\Xi^{*}(\pi) &= \sup_{u\in E} \left \{ \int ud\pi - \int \varphi d \mu- \int \psi d\nu  \,\middle\vert\,  u = \varphi \oplus \psi \text{ for some } (\varphi, \psi) \in L^{1}(d \mu) \times L^{1}(d \nu) \right \} \\
&= \begin{cases}
0  &\text {if } \quad  \forall(\varphi, \psi) \in L^{1}(d \mu) \times L^{1}(d \nu) : \int \varphi \oplus \psi d \pi= \int \varphi d \mu+\int \psi d \nu \\
 +\infty & \text {else}
\end{cases} \\
&= \begin{cases}
0  &\text {if } \quad  \pi \in \Pi(\mu, \nu) \\
 +\infty & \text {else}.
\end{cases}
\end{align*}
$$

It follows that
$$
\max _{\pi \in E^{*}}\left[-\Theta^{*}\left(-\pi\right)-\Xi^{*}\left(\pi\right)\right] = \max _{\pi \in \Pi(\mu, \nu) \cap M_{+}(X \times Y)} -\int c d \pi = - \min _{\pi \in \Pi(\mu, \nu)} \int c d \pi.
$$
Hence
$$
-\sup \left\{J(\varphi, \psi) \mid (\varphi, \psi) \in \Phi_{c}\right\} = - \min _{\pi \in \Pi(\mu, \nu)} \int c d \pi.
$$