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I refer to notes about entropy-regularized optimal transport, at https://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf

In Theorem 3.2, it says that the Schrodinger potentials achieve the optimum of the entropy-minimization problem, with: $H(\pi_* | R) = \inf_{x \in \Pi(\mu, \nu)} H(\pi | R) = \mu(\phi_*) + \nu(\psi_*)$.

I understand that both $\mu, \nu$ are measures over spaces $X, Y$. In my case, both input spaces are $\mathbb{R}^n$ and $\mu, \nu$ are probability measures, so we have $\mu: X = \mathbb{R}^{n} \rightarrow \mathbb{R}$; $\nu: Y = \mathbb{R}^{n} \rightarrow \mathbb{R}$; and that integrations over the whole space should sum to one.

Schrodinger potentials $\phi, \psi$ are defined that satisfy $\phi : X \rightarrow \mathbb{R}$, $\psi : Y \rightarrow \mathbb{R}$.

I don't understand how to evaluate the two functions at the Schrodinger potentials: $\mu(\phi_*) + \nu(\psi_*)$. It seems like $\phi_*$ is not in the input domain of $\mu$? Can anyone explain the above expression $\mu(\phi_*) + \nu(\psi_*)$ for me, in real spaces $\mathbb{R}^n$? Does it refer to some integral e.g. $\int \phi_*(x) \mu(x) \cdots \mathrm{d} x$?

I am unfamiliar with measure theory, so any help would be greatly appreciated. Thank you!

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  • $\begingroup$ Yes, simply integrals. $\endgroup$
    – Tobsn
    Jun 23 at 21:42
  • $\begingroup$ Can you write out the expression for me? Thank you! $\endgroup$ Jun 23 at 21:43
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    $\begingroup$ $\int\phi_{*}d\mu$ $\endgroup$
    – Tobsn
    Jun 23 at 21:44
  • $\begingroup$ I see. In real spaces, does it correspond to something like: $\int \phi_*(x) \mathrm{d} \mu(x) = \int \phi_*(x) \mu(x) \mathrm{d} x$? $\endgroup$ Jun 23 at 21:45
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    $\begingroup$ The identity would be true only provided $\mu$ has Lebesgue-density. $\endgroup$
    – Tobsn
    Jun 23 at 21:49

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