Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that $$ \int\limits_E \phi(x,y)dy=1 $$ for all $x\in E$. Let us consider the non-increasing sequence of non-empty compact sets $A_n$ such that for all $x\in A_{n+1}$ we have $$ \int\limits_{A_n} \phi(x,y)dy=1. $$ Since $A_n$ are compacts, there exists a non-empty limit set $A = \bigcap\limits_n A_n$

Do we have for all $x\in A$ that $$ \int\limits_A \phi(x,y)dy = 1? $$

thinkthis might mean is this: think of $\phi$ as the transition kernel for a MC. Let $A_0=E$; Let $A_1$ be the support of the random variable $X_1$ given that $X_0$ has some distribution with support $E$. Now let $A_2$ be the support of $X_2$ etc. Is this right? – Anthony Quas Jan 23 '11 at 17:24