The Lebesgue decomposition theorem says that any $\sigma$-finite measure can be decomposed into the sum of:
- an absolutely continuous measure,
- a singular continuous measure,
- and a discrete measure.
I'm wondering if the following measure could possibly have a singular continuous component.
Let $M$ be an $(n-1)$-dimensional embedded $C^1$ submanifold of $\mathbb R^n$. Let $\mu$ be the measure on the projective space $\mathbf P(\mathbb R^n)$ defined by
$$\mu(A) = \mathcal H_{n-1}(\{t\in M : N_s M \subseteq A\}),\qquad A \subseteq \mathbf P(\mathbb R^n),$$
where $H_{n-1}$ denotes the Hausdorff measure, and $N_s M$, the normal space of $M$ at $s$.
For sure, $M$ can be constructed such that $\mu$ has atoms corresponding to the "flat" parts of $M$. How can I relate the smoothness of $M$ to the properties of the measure $\mu$?