# Fixed point: sets and measures

Let $X$ be a Borel space with a Borel measure $\mu$. Suppose $\xi: X\times X\to\mathbb R_{\geq 0}$ is a continuous function and put $s(x) = \{y\in X:\xi(x,y) = 0\}$. For any set $b\in\mathcal B(X)$ we put $$\mathcal Sb = \bigcup\limits_{x\in b}s(x).$$ I am interested in the solutions of an equation $\mathcal Sb\subseteq b$, or even the generalized one: $$\mu(\mathcal Sb\setminus b) = 0.$$ Could we say that these equations are fixpoint problems? If there is a literature for such problems?

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When you say solution, do you mean you are solving for $b$ where $\xi$ is given? – Willie Wong Jan 18 '12 at 10:50
The statement of your problem strikes me as somewhat similar to van Maaren's version of Sperner lemma, and I would agree if you choose to call the equations fixed point problems. (Though I don't see how one can reduce your problem down to finite combinatorics, so I am not exactly sure where one would find a result of this sort.) – Willie Wong Jan 18 '12 at 10:58
@Willie: I happen to work in the same university as Prof. van Maaren. Could you give me a reference for his version of Sperner lemma? – Ilya Jan 18 '12 at 12:29
The original article is Generalized pivoting and coalitions which appeared in the book "The Computation and Modelling of Economic Equilibria". A very nice write-up about it (and some related consequences) can be found in van de Vel's Theory of Convex Structures, Chapter IV section 6.9 et seq. – Willie Wong Jan 18 '12 at 14:05
@Willie: thank you very much – Ilya Jan 18 '12 at 20:28

The following is not so much an answer as it is a pointer to some connections with other areas that one might exploit.

Assume for simplicity that $X$ is a compact metric space. All the results I use can be found in the book Infinite Dimensional Analysis by Aliprantis and Border.

Then every closed subset of $X\times X$ is the zero-set of some function $\xi$ (just let $\xi$ be the distance to the subset), so we can dispense with the function $\xi$. This closed subset is the graph of the correspondence (or multifunction) $s$. That the graph is closed is equivalent to $s$ being an upper hemicontinuous correspondence (uhc) with closed values. Then $\mathcal{S}$ is simply the usual forward image of a set under a correspondence. The coordinatewise union of two uhc correspondences is uhc, so $s^*$ given by $s^*(x)=s(x)\cup\{x\}$ is uhc and compact-valued.

You are looking for the Borel sets $B$ such that $s^*(B)=B$. A compact valued uhc correspondence maps compact sets to compact sets, so $s^*(B)$ is an increasing function on the complete lattice of compact sets in $X$. So the set of compact fixed points forms a complete lattice, by the Tarski fixed point theorem. It might, however, be the case that $\emptyset$ and $X$ are the only such fixed points.

For general Borel sets, the problem is that uncountable unions of Borel sets are usually not Borel, so $s^*$ doesn't map $\mathcal{B}(X)$ to itself.

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"For general Borel sets, the problem is that uncountable unions of Borel sets are usually not Borel" That, I think, is the crux of the difficulty. – Willie Wong Jan 18 '12 at 14:09
+1 Michael, thanks a lot for the answer. I hope you can excuse me if I write a comment in a couple of days - I have to figure out something w.r.t. this problem. – Ilya Jan 18 '12 at 20:29