Nonlinear harmonic measures on trees

We show that nonlinear harmonic measures on trees lack many desirable properties of set functions encountered in classical analysis. Let F be an averaging operator on R^K and W^F be the F-harmonic measure on a K-regular forward branching tree. Unless F is the usual average, W ^F is not a Choquet capacity; union of sets of W^F measure zero can have positive W^F measure when F is permutation invariant; and there exist sets of full W^P measure having "small" dimension. Let A be a monotone operator on R^K, then A -harmonic functions on trees need not obey the strong maximum principle unless the ratio of the ellipticity constants is close to 1.