Let $(M,\phi)$ be a Finsler manifold. The following definition is taken from Wikipedia.
Let $A\subseteq M$ be a measurable set, then the $n$-dimensional Holmes-Thompson volume is defined by \begin{equation} \operatorname{vol}_{n}^{HT}(A,\phi):=\int_{B^{*}A}\frac{\omega ^n}{n!}, \end{equation} where $\omega$ is the standard symplectic form on $T^*M$ and $B^{*}A$ is the set of tuples $(p,v)\in T^*M$, such that $p\in A$ and $\|v\|\leq 1$.
What does it mean for the set $A\subseteq M$ to be measurable?
Thanks for your help!
The Finsler $\phi:TM\to\mathbb{R}_{\geq0}$ defines a metric $d_\phi$ on $M$. $d_\phi$ defines a topology $\mathcal{T}_{d_\phi}$ on $M$. $\mathcal{T}_{d_\phi}$ defines a $\sigma$-algebra $\mathcal{B}_{\mathcal{T}_{d_\phi}}$ (= smallest collection of subsets containing $\mathcal{T}_{d_\phi}$ and closed under intersections and countable unions) on $M$. Most likely $A\subseteq M$ being measurable means $A \in \mathcal{B}_{\mathcal{T}_{d_\phi}}$.