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 sequence of 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 limit set $A = \lim_n A_n$. Do we have for all $x\in A$ that $$ \int\limits_A \phi(x,y)dy = 1? $$

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 $$ \int\limits_A \phi(x,y)dy = 1? $$