I have the following definition of $\alpha$-dimensional Hasudorff measure
Let $E$ be subset of $\mathbb{R^n}$. Fix $\alpha \in [0, \infty)$, $\kappa \in \mathbb{R}^+$ and $\epsilon > 0$. Then $\alpha$-dimensional Hausdorff measure (outer) is defined as: \begin{align} H_{\alpha}(E) : = \displaystyle \lim_{\epsilon \to 0} \; H^{{\epsilon}}_{\alpha}(E) = \sup_{\epsilon > 0} H^{{\epsilon}}_{\alpha}(E) \end{align}
Where for every $\epsilon>0$, define \begin{align*} H^{{\epsilon}}_{\alpha}(E) := \inf \left\{\sum_{j\geq1} \kappa r^{\alpha}_{j} : E \subset \bigcup_{j \geq 1} B(x_j, r_j) \text{ with } r_j < \epsilon \; \forall j \right\} \end{align*} where infimum is taken over all such $\epsilon$-covers of $E$.
$B(x_j, r_j)$ is ball in $\mathbb{R^n}$ under usual Euclidean metric.
I want to know what is the relation between Hausdorff measure of $E$ and Lebesgue measure of $E$. Also when does Hausdorff measure coincides with Lebesgue measure?