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 · and!F be the F-harmonic measure on a ·-regular forward branching tree. Unless F is the usual average,!F is not a Choquet capacity; union of sets of!F measure zero can have positive!F measure when F is permutation invariant; and there exist sets of full!F measure having "small" dimension. Let A be a monotone operator on R · , then A-harmonic functions on trees need not obey the strong maximum principle unless the ratio of the ellipticity constants is close to 1