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We show that if the upper Assouad dimension of the compact set $E\subseteq\mathbb{R}$ is positive, then given any $D>\dim_{A}E$ there is a measure withsupport $E$ and upper Assouad (or regularity) dimension $D$. Similarly, givenany $0\leq d<\dim_{L}E,$ there is a measure on $E$ with lower Assouad dimension$d$.

Measures with specified support and arbitrary Assouad dimensions