For any point $x \in \mathbb R^n$, denote by $\delta\_x$ the Dirac Delta measure centered at $x$.


Let $a\_n$ be an sequence of positive numbers with $\lim\_{n \to \infty} a\_n = 0$, and let $d\_i$ be a countable dense subset of $\mathbb R^n$.


Given a rearrangement $a\_{n\_k}$ of $a\_n$, consider the space $\mathbb R^n$ under the Euclidean topology and the measure $\sum\_k a\_{n\_k} \delta\_{d\_k}$.



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> **Question:** For any given $a\_n, d\_i$ satisfying the above consitions, does there exist a rearrangement such that the above space admits nowhere locally constant continuous functions that are integrable under the given measure?
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*Note: A function is said to be nowhere locally constant if there exists no open set $S$ for which $f(x) = c$ for all $x$ in $S$.*



Does $\mathbb R^n$ equipped with a sum of Dirac delta measures admit nowhere locally constant continuous integrable functions?
Partition $(a\_n)$ into two subsequences $(a\_n')$ and $(a\_n'')$ with $\sum a\_n' < \infty$, and partition $(d\_i)$ into two subsequences $(d\_i')$ and $(d\_i'')$ such that $d\_i'' \to \infty$. Pair the $a\_i'$s with the $d\_i'$s and the $a\_i''$s with the $d\_i''$s. Then any bounded continuous function that vanishes at every $d\_i''$ will be integrable.


