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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.

roi_saumon
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  • The second part immediately follows from the relations between $\phi$ and $\phi^{c}$ (bottom p. 10) and the integrability assumptions on the cost. – Tobsn Jul 06 '21 at 18:26
  • $\phi^{c+}\le c(x,y)-\phi(x)$ so $\int\phi^{c+}d\gamma\le \int c(x,y)d\gamma-\int\phi(x)d\gamma$ so $\int \phi^{c+}d\nu\le\int cd\gamma-\int\phi d\mu$ but now how do we handle $\max{\phi^{c+},0}$? – roi_saumon Jul 06 '21 at 22:59

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