  In this note we show that locally $p$-admissible measures on $\mathbb{R}$
necessarily come from local Muckenhoupt $A_p$ weights. In the proof we employ
the corresponding characterization of global $p$-admissible measures on
$\mathbb{R}$ in terms of global $A_p$ weights due to Bj\"orn, Buckley and
Keith, together with tools from analysis in metric spaces, more specifically
preservation of the doubling condition and Poincar\'e inequalities under
flattening, due to Durand-Cartagena and Li. As a consequence, the class of
locally $p$-admissible weights on $\mathbb{R}$ is invariant under addition and
satisfies the lattice property. We also show that measures that are
$p$-admissible on an interval can be partially extended by periodical
reflections to global $p$-admissible measures. Surprisingly, the
$p$-admissibility has to hold on a larger interval than the reflected one, and
an example shows that this is necessary.
