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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 $S$.

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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.

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  • $\begingroup$ Ah, this was much easier than I expected. Slick solution though! $\endgroup$
    – Nate River
    Sep 28, 2021 at 4:22
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    $\begingroup$ Thanks! Fun question. $\endgroup$
    – Nik Weaver
    Sep 28, 2021 at 4:23

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