Couplings as generalized functions

I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures $$\mu$$ and $$\nu$$ on a metric space $$X$$ by $$W_p(\mu, \nu) = \left(\inf_\gamma \int_{X\times X} \mathrm{dist}(x,y)^p \mathrm{d}\gamma(x,y)\right)^{1/p},$$ where the infimum is taken over all measures $$\gamma$$ on $$X\times X$$ whose projections onto the respective factors give $$\mu$$ and $$\nu$$.

It occurred to me that this appears to provide a framework to generalize the notion of functions altogether. In particular, for a measure space $$X,\mu$$, we could define a generalized function from $$X$$ to $$Y$$ as a measure $$\gamma$$ on $$X\times Y$$ such that $$\pi_* \gamma = \mu$$, where $$\pi:X\times Y\rightarrow X$$ is the projection to the first factor.

We can then try to define all the usual operations on functions in this new framework. For instance, for generalized funcitons $$\gamma_1$$ from $$X$$ to $$Y$$ and $$\gamma_2$$ from $$Y$$ to $$Z$$, we could try to define $$\gamma_2\circ \gamma_1$$ by $$\gamma_2\circ \gamma_1 (A\times C) = \sum_{y\in Y}\gamma_1(A\times \{y\})\gamma_2(\{y\}\times C).$$ Of course, this only works if $$Y$$ is finite (or countable). I believe there should be a generalization of this to arbitrary $$Y$$ using integrals, but I am having trouble coming up with a good formulation. Apparently you can use Markov chains, but I am still trying to understand how to write this down cleanly.

In any case, I am interested in whether this notion provides a useful way to generalize functions. How much of the usual theory carries over? Could this be used e.g. to study weak solutions to PDEs, to study the monodromy of solutions to algebraic equations, or to study backwards iteration in otherwise non-invertible dynamical systems?

• How is a (measurable?) map $f:X \to Y$ a particular case of your proposed definition? Nov 3 '21 at 18:04
• We would take $\gamma(E) = \mu_X(\pi_1(E \cap \Gamma_f))$, where $\Gamma_f \subset X\times Y$ is the graph of $f$. In particular, $\gamma(A \times B) = \mu_X(A \cap f^{-1}(B))$. Nov 3 '21 at 18:20
• You've got a serious problem here: who guarantees that $\pi (E \cap \Gamma_f)$ is measurable? Remember that measurable maps don't always take measurable subsets into measurable subsets; there are cases when this is true, but they are very restrictive (they require $\pi$ to be injective, among other conditions). So your formula does not define a measure $\gamma$. And remember, the natural operation on measures is the pull-back, not the push-forward. So I am very pessimistic about your project; I believe that it is hopeless, save for some trivial cases. Nov 3 '21 at 20:40
• @AlexM. $\pi$ restricted to the graph of $f$ is the inverse of the measurable map $(\mathrm{id} \times f):X \rightarrow X\times Y$. If it’s more clear, just go by the second version I wrote, which will clearly always be well-defined for $f$ measurable. Nov 3 '21 at 20:44
• @AlexM Are you sure you don't confuse pull-back and push-forward? Measures are covariant!
– R W
Nov 4 '21 at 3:19

Since you refer to transportation metrics, it seems to be fair to assume that your probability spaces are Lebesgue (=standard). Then it is the same to talk either about "lifting" your measure $$\mu$$ to $$\gamma$$ or about the family of the conditional measures of the projection $$\gamma\to\mu$$ (or, which is also the same, about the corresponding Markov kernel). Ordinary functions from $$X$$ to $$Y$$ correspond then to the situation when all transition measures are delta-measures.

This construction is known as mutivalued maps with invariant measure or polymorphisms (the term introduced by Vershik, also see Schmidt - Vershik or Neretin).

• This is exactly what I was looking for -- thank you! Nov 30 '21 at 9:54

$$\newcommand\ga\gamma\newcommand\ze\zeta$$

1. First of all, your operation $$\circ$$ has hardly anything to do with the Wasserstein metric. It is just an operation over measures over product spaces.

2. Assuming that $$\ga_1$$ is a probability measure over a product space $$X\times Y$$ (endowed with a corresponding product $$\sigma$$-algebra) and $$\ga_2$$ is a probability measure over a product space $$Y\times Z$$, let $$(\xi,\eta)$$ and $$(\eta',\ze)$$ be independent random elements of $$X\times Y$$ and $$Y\times Z$$, respectively (defined on the same probability space $$(\Omega,\mathcal F,\mathsf P)$$), with the respective distributions $$\ga_1$$ and $$\ga_2$$. Then for all appropriately measurable sets $$A\subseteq X$$ and $$C\subseteq Z$$, $$(\ga_2\circ\ga_1)(A\times C)=\mathsf P(\xi\in A,\eta=\eta',\zeta\in C). \tag{1}$$ In my long career in probability, I have never seen probabilities like this, nor can I envision any use of them.

3. It is obvious from (1) that the measure $$\ga_2\circ\ga_1$$ will be a probability measure only in the trivial case when $$P(\eta=\eta'=c)=1$$ for some $$c\in Y$$.

4. The "composition" $$\ga_2\circ\ga_1$$ will be very unstable with respect to the distributions of the random elements $$\eta$$ and $$\eta'$$ of $$Y$$ -- a slight change in the distribution of either one of these two random elements may change the value of $$(\ga_2\circ\ga_1)(A\times C)$$ by as much as $$1$$.

5. As stated in the question you linked to your post, one can and does compose transition probabilities $$P(x,B)$$ and $$Q(y,C)$$: $$(QP)(x,C)=\int_Y P(x,dy)Q(y,C).$$ This does generalize the usual composition of functions. Indeed, the composition of (appropriately measurable) functions $$f\colon X\to Y$$ and $$g\colon Y\to Z$$ corresponds to the case when $$P(x,B)=1(f(x)\in B)$$ and $$Q(y,C)=1(g(y)\in C)$$, so that $$(QP)(x,C)=1((g\circ f)(x)\in C)$$.