Crofton's formula can be expressed as $$2L=\int_{\cal L} {\cal H}^0(C\cap \ell) d\ell,$$ where $L$ is the length of a smooth planar curve $C$, $\cal L$ is the set of all lines in the plane, and ${\cal H}^0(C\cap \ell)$ counts the number of intersection of $C$ and a line $\ell\in{\cal L}$. Notice that ${\cal L}$ is a two-dimensional space, so the measure on it is something like ${\cal H}^2$.
I'm interested in an integral similar to the one appearing in this formula, namely $$\int_{{\cal L}_\text{tan}} {\cal H}^0(C\cap \ell) d\ell,$$ where here one is integrating over ${\cal L}_\text{tan}$, which consists of those lines that are tangent to the curve $C$ and the measure on this space is something like ${\cal H}^1$ as this is a generally a one-dimensional space.
Is this integral finite? It seems like this is a question in integral geometry, but I'm not very familiar with the subject.