I am trying to read Ambrosio's user guide to optimal transport and I am having some difficulties with the following Theorem
Theorem 1.13 (Fundamental theorem of optimal transport) : Assume that $c : X × Y → R$ is continuous and bounded from below and let $μ ∈\mathcal{P}(X)$, $ν ∈ \mathcal{P}(Y )$ be such that $c(x, y) ≤ a(x) +b(y)$, for some $a ∈ L ^1 (μ), b ∈ L^1 (ν)$. Also, let $γ ∈ Adm(μ,ν)$. Then the following three are equivalent:
- the plan $γ$ is optimal,
- the set $\operatorname{supp}(γ)$ is $c$-cyclically monotone,
- there exists a $c$-concave function $φ$ such that $\max\{φ, 0\} ∈ L^1 (μ)$ and $\operatorname{supp}(γ) ⊂ ∂^{c+} φ$.
First, I was wondering why it was important that $\max\{\varphi,0\}\in L^1(\mu)$. Is it just important that the positive part of this function is integrable? Second, it is later stated (remark 1.15) that $\max\{\varphi,0\}\in L^1(\mu)$ implies that $\max\{\varphi^{c+},0\}\in L^1(\nu)$ but I am not sure how this is true.