Reduced function

Summary

Let $u$ be a positive superharmonic function and $E$ a set. The reduced function, or réduite, of $u$ on $E$ is the lower envelope of all positive superharmonic functions that dominate $u$ on $E$. Its lower semicontinuous regularisation is again superharmonic and equals $u$ on $E$ up to a polar set. When $u$ is the potential of a measure, this regularisation is the potential of the balayage of that measure onto $E$. Reduced functions give capacities, harmonic measures and thinness in one framework.