Iterated fine limits

Abstract:

Let $v$ and $u$ be, respectively $n$-superharmonic and $n$-harmonic functions on the product of $n$ harmonic spaces. We prove that the iterated fine limits of $\frac {v}{u}$ exist and are independent of the order, for $\lambda$ almost every minimal boundary element where $\lambda$ represents the function $u$. As an application we prove an important property concerning the reduced function of a positive $n$-harmonic function.