Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus.  It's known that $C_p$ has Hausdorff dimension $\le n-1$.  $C_p$ may not be bounded; for example, consider the cylinder.  For $R>0$, consider the subset 
$$
C_{p,R}=C_p\cap \overline{B(p,R)}.
$$
Let $A_\delta$ be the $\delta$-neighborhood of $C_{p,R}$, i.e.
$$
A_\delta=\{x\in M, d(x,C_{p,R})<\delta\}.
$$
My question is: For any $\epsilon>0$,  is there $\delta>0$, such that the n dimensional Hausdorff measure (it coincides with Riemannian volume for a Riemannian manifold)
$$
H^n(A_\delta)<\epsilon?
$$