Let $\mathcal{H}^k$ be the $k-$dimensional Hausdorff measure on $\mathbb R^n$, with $k \in \{1, \ldots n\}$. By Carathéodory's theorem we know that there exists a sigma algebra $\mu(\mathcal{H}^k)$ of $\mathcal{H}^k-$measurable sets and it's posible to show that if $f:\mathbb R^k \to \mathbb R^n$ is Lipschitz and $A \subset \mathbb R^k$ is a Lebesgue measurable set of $\mathbb R^k$ then $f(A) \in \mu(\mathcal{H}^k)$ my questions are:

**Is the converse of the last assertion true?**

**There exists a characterization of the sets in $\mu(\mathcal{H}^k)$ ?**

**What relations exist between the Lebesgue sigma algebra on $\mathbb R^n$ and $\mu(\mathcal{H}^k)$?**

For the third question using the identity function $I:\mathbb R^n \to \mathbb R^n$ which is Lipschitz, its easy to see that if $A$ is a Lebesgue measurable subset of $\mathbb R^n$ then $A=I(A) \in \mu(\mathcal{H}^n)$, then in this case the Lebesgue sigma algebra is contained in $\mu(\mathcal{H}^n)$ this contention is proper?