I am working with an absolute continuous curve $c:[a,b]\rightarrow M$ on a smooth manifold $M$. The manifold is equipped with a metric $d$ and the curve is absolute continuous with respect to $d$.
For $M=R^n$ equipped with the euclidean metric it is well known that the derivative $c':[a,b]\rightarrow R^n$ is measurable.
I am wondering if I can prove that $\dot{c}:[a,b]\rightarrow TM$ is measurable. Besides, how would I define measurability on $TM$.