Integration of H\"older forms and currents in snowflake spaces

For an oriented $n$-dimensional Lipschitz manifold $M$ we give meaning to the integral $\int_M f dg_1 \wedge ... \wedge dg_n$ in case the functions $f, g_1, >..., g_n$ are merely H\"older continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for $\alpha \in (\frac{n}{n+1},1]$ the $n$-dimensional locally normal currents in a locally compact metric space $(X,d)$ represent a subspace of the $n$-dimensional currents in $(X,d^\alpha)$. On the other hand, for $n \geq 1$ and $\alpha \leq \frac{n}{n+1}$ the latter space consists of the zero functional only.