Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.

For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff dimension $\beta$?

In case this is true, could you provide a reference for this statement?

Added: Actually I am happy if $A$ is compact.