Let $(M,g)$ be a Riemannian symmetric space and $f:M\to\mathbb{R}$ be a non-negative integrable function. Fix a geodesic $\alpha(t)$. Define
$$
U(t):=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut locus ${\rm Cut}(\alpha(s))$ of $\alpha(s)$, $s\in[0,t]$. Questions:

(1) Is $U(t)$ a measurable set? 

(2) Is the following quantity finite:
$$
{\lim\sup}_{t\to 0}\frac{\int_{U(t)}f(x) {\rm dvol}(x)}{t}.
$$