Inner regularity on open sets implies inner regularity on all Borel sets

These sets define a Hausdorff topology: wedges with different centres become disjoint after truncation, and isolated points can be removed by truncation. Each $W_N(y)$ is compact, since any neighbourhood of its centre leaves only finitely many of its atoms uncovered. Thus $X$ is locally compact. The axis is closed and discrete; each subset of it is closed in $X$, so every subset of $X$ is Borel. At level $n$ a wedge contains at most $2n+1$ atoms, giving $\mu (W_N(y))\le {\sum }_{n>N}(2n+1)n^{-3}\to 0$. In particular $\mu$ is locally finite.

For a subset $S\subseteq A$, if an open $U\supseteq S$ has finite $\mu$-mass, deleting finitely many atoms from $U$ makes its mass arbitrarily small without losing $S$. Hence $M(S)$ is either zero or infinity. The family $\mathcal{I}=\{S\subseteq A:M(S)=0\}$ is closed under subsets and countable unions: cover its $j$th member by an open set of mass less than $\epsilon 2^{-j-1}$, for $j\ge 0$, and take their union. Thus $\nu (S)=0$ on $\mathcal{I}$ and $\nu (S)=\mathrm{\infty }$ otherwise is a countably additive measure on the axis.

Let $U\supseteq A$ be open and set $B_N=\{y\in [1,2]:W_N(y)\subseteq U\}$. Each $B_N$ is closed in $[1,2]$: its complement is the union, over missing atoms with $n>N$, of the open intervals $\mathrm{\mid }m\mathrm{/}{n}^2-y\mathrm{\mid }<1\mathrm{/}n$. The $B_N$ increase and cover $[1,2]$. By Baire some $B_N$ contains a nondegenerate interval $I\subseteq [1,2]$. All atoms with $n>N$ and $m\mathrm{/}{n}^2\in I$ belong to $U$, so $\mu (U)\ge {\sum }_{n>N}(\mathrm{\mid }I\mathrm{\mid }{n}^2+O(1))n^{-3}=\mathrm{\infty }$. Hence $M(A)=\mathrm{\infty }$.

For every $E\subseteq X$ one has $M(E)=\mu (E\cap D)+\nu (E\cap A)$. Indeed the lower bounds follow by monotonicity; when $\nu (E\cap A)=0$, adjoin an arbitrarily small open cover of $E\cap A$ to the open set $E\cap D$. The infinite cases follow directly from the lower bounds. Consequently $M$ is a Borel measure. For open $U$, $M(U)=\mu (U)$ by the defining infimum. The same infimum makes $M$ outer regular.

Every compact set has finite $M$-mass by a finite cover of finite-mass basic neighbourhoods. Also $M(\{d\})=\mu (\{d\})$ for $d\in D$, so finite sets of atoms inside an open $U$ have masses with supremum $\mu (U)=M(U)$. This proves compact inner regularity on opens. A compact subset of the closed discrete axis is finite, and each axis singleton has $M$-mass zero by the wedge estimate; therefore every compact subset of $A$ has mass zero.

We have $M(A)=\mathrm{\infty }>0={\sup }_{K\subseteq A\text{ compact}}M(K)$, although $M$ satisfies all the claimed premises. This refutes the implication, preserving the page's distinction between Radon and all-Borel regularity.