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 $\sum_k a_{n_k} \delta_{d_k}$.

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?

*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 $S$.*