PomeProposition 4.2.5.9E787ECDB068 Let $N$ be a fine neighborhood of a point $x\in \Omega $. Then every Euclidean ball centered at $x$ contains a Borel subset of $N$ of positive Lebesgue measure. In particular, the fine topology has no isolated points.
Proposition 4.2.59E787ECDB068
Proof. By proposition 4.2.4, $N$ contains a set $\{ y\in V:u(y)>a\} $ with $u\in \mathrm{SH}(V)$ and $u(x)>a$. It suffices to consider balls $B(x,r)$ with $\overline{B(x,r)}\subset V$. The set $E=\{ y\in B(x,r):u(y)>a\} $ is a Borel subset of $N$, because $u$ is Borel measurable. Let $M$ be a finite upper bound for $u$ on $\overline{B(x,r)}$; then $M\geq u(x)>a$. Since $u\leq M$ on $E$ and $u\leq a$ on $B(x,r)\setminus E$, the ball submean inequality of theorem 3.4.5 gives
\[ u(x)\leq \frac{1}{|B(x,r)|}\int _{B(x,r)}u(y)\, \mathrm d y\leq a+(M-a)\frac{|E|}{|B(x,r)|}. \]
Hence $|E|\geq |B(x,r)|\, (u(x)-a)/(M-a)>0$. As a singleton has Lebesgue measure zero, it is not a fine neighborhood of its point. $\square$
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