# Does $\mathbb R^n$ equipped with a sum of Dirac delta measures admit nowhere locally constant continuous integrable functions?

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 $$S$$.

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.