Optimal objective of Static Schrodinger Bridge

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!

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