Interval integrability of the running cost along an admissible pair

Let $g$ be a continuous running cost on state-control pairs and let $(u,x)$ be admissible on $[t_0,T]$ from the state $\xi$ in the sense of BertsekasCTAdmissibleFrom: $u$ has bounded image on $[t_0,T]$ and is continuous off a finite set, and $x$ is continuous on $[t_0,T]$.

Then for any two times $a,b\in[t_0,T]$ the running cost is interval integrable:

The integrand is continuous off a finite set, hence almost everywhere continuous and measurable, and it is bounded because $x([t_0,T])$ is compact, $u([t_0,T])$ is bounded, and $g$ is continuous on the resulting compact product. This is the fact that makes the cost functional $h(x(T))+\int_{t_0}^{T} g(x(t),u(t))\,dt$ of an admissible pair a genuine Lebesgue integral, and it is what allows the cost to be split at intermediate times.