Finely Harmonic Functions need not be Quasi-Analytic

If a function f is harmonic on a connected open set R^d andis constant in a neighbourhood of one point in then it is identicallyconstant. We give an example of a non-constant finely harmonicfunction defined on E={zC||z|<2}p which is identically zeroon |z|1. The exceptional set P is of capacity zero so E is finelyopen and finely connected. This example therefore shows thatfinely harmonic functions are not globally determined by theirlocal behaviour.