Invariant integration on pseudo-Finslerian manifolds

We consider n-dimensional oriented pseudo-Finslerian manifolds with mth root metrics. Natural volume forms on these manifolds are defined. The natural volume forms depend on the parity of the positive integer m>1 and are expressed in terms of Cayley hyperdeterminants. With the help of the introduced natural volume forms, we define invariant integrals of finite functions over n-dimensional oriented pseudo-Finslerian manifolds with mth root metrics.