Measurability of the union of cut loci along a curve

Question

Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define $$ U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s)) $$ as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\alpha$. I am curious whether the set $U(t)$ is measurable or not. One may assume $t$ is small whenever necessary.