TITLE: Existence of subset with given Hausdorff dimension
QUESTION [22 upvotes]: 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.

REPLY [2 votes]: The following is Corollary 7 of [1].
Theorem: For $X$ (an analytic subset of) a complete separable metric space, and $ s \in [0,\infty)$, the following is true about the Hausdorff measure $\mathcal{H}^s$: For every $ l < \mathcal{H}^s(X) $ there exists a compact subset $A \subset X$ such that 
$$
l < \mathcal{H}^s (A) < \infty \, .
$$
[1] J. D. Howroyd, On dimension and on the existence of sets of finite positive Hausdorff measure, 1993