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?
Fixed point: sets and measures
7
$\begingroup$
reference-request
measure-theory
set-theory
fixed-point-theorems
-
0When you say solution, do you mean you are solving for $b$ where $\xi$ is given? – 2012-01-18
-
1The 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.) – 2012-01-18
-
0@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? – 2012-01-18
-
0The 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. – 2012-01-18
-
0@Willie: thank you very much – 2012-01-18